Keywords

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

2.1 Structures

Carbon is the sixth element of the periodic table with the ground state level 3P0 and electron configuration 1s 2 2s 2 2p 2, which is hybridized to form sp 1, sp 2 and sp 3 chemical bonds between atoms. The general oxidation states (numbers) of carbon in numerous inorganic compounds are (−4) (+4) and (+2); the radii of carbon are:

  • atomic (metallic, CN = 12)—0.091 nm,

  • atomic (van der Waals)—0.170 nm,

  • atomic (covalent)—various, see Table 2.1,

    Table 2.1 Comparative atomic bonding characteristics of s- and p-elements of the periodic table [2–9]
    Full size table

  • ionic (+4)—0.015 nm (CN = 4) or 0.016 nm (CN = 6);

its electronegativity is 2.55 in Pauling scale, or 2.50 in Allred–Rochow scale. The hybridized sp n-bonding (1 ≤ n ≤ 3) differs considerably from common s- or p-bonds inherent to other chemical elements of the basic periods. The values of energy characteristics, in particular, specific bond energies (average bond energy per bond order) of carbon–carbon sp n-bonds occupy a wide area of intermediate positions between those characteristics of s- and p-elements, depending substantially on the hybridisation type of bonding: s > sp 3 > sp 2 > sp 1 > p (Table 2.1). The electron hybridization predetermines the ability of carbon atoms to arrange numerous compounds with other elements of the periodic table, but also the various allotropes, such as carbyne/carbyte, graphene/graphite and diamond (Table 2.2), which are structured with linear, planar and tetrahedral symmetry, coordination numbers of 2, 3 and 4 and the bond angles of 180°, 120° and 109.47°, respectively for sp 1-, sp 2- and sp 3-hybridized forms. According to Mendeleev “none of the elements can compete with carbon in ability to form complex structures” [1, 540].

Table 2.2 Systematic classification of carbon nano- and macrostructures
Full size table

There has been a new wave of interest in carbon materials caused by the recent discoveries in the nanoscience [2, 9–14, 31, 42, 45–53], which has affected all science and engineering fields. Some points of view are developed in literature to classify carbon macro- and nanostructures. In this way, schemes based on hybridization characteristics were suggested by McEnaney [9], Heimann et al. [10], Inagaki [11], Belenkov [12], Shenderova et al. [13, 78] and Falcao and Wudl [14]. The systematic classification of all the experimentally confirmed and theoretically predicted carbon structures, which is given in Table 2.2, is based mainly on the relative values of the specific bond energies of sp n-hybridized carbons as well as carbon phase transformation diagram (see Sect. 2.2). This approach, employing within the overall hierarchy of classified carbon structures from submolecular to macroscopic scales, in accordance to the bonding energy criteria, allows marking out among the variety of carbon structures three general families (with integer degree of hybridisation index n):

  • carbyne general family (n = 1), including sp 1-hybridized (σ-sp 1 + 2π) carbons from nanostructured carbyne to its macrostructural crystalline form “carbyte” (the existence of carbyne (or carbolite) has not been universally accepted; the proposed term for a macrostructural form is given on the basis of analogy with graphene–graphite and fullerene–fullerite correlations);

  • graphene general family (n = 2), including sp 2-hybridized (σ-sp 2 + π) carbons from nanostructured graphene to macrostructural crystalline graphite with its polytypic (hexagonal and rhombohedral) forms;

  • diamond general family (n = 3), including sp 3-hybridized (σ-sp 3) carbons from various nanostructured diamonds to macrostructural crystalline diamond with its polytypic (cubic and hexagonal) forms.

and two intermediate (or transitional) families (with non-integer degree of hybridisation index n):

graphyne intermediate family (1 < n < 2), including sp 1<n<2-hybridized (σ-sp 1<n<2 + (3 − n)π) carbons from nanostructured graphyne (analogue of graphene modified by triple bonds) to macrostructural “graphyte” with various polytypes, which are characterized by the intermediate between carbyne and graphene hybridisation type (currently, hypothetical structures calculated and predicted only theoretically; the proposed term for a macrostructural form is given on the basis of analogy with graphene–graphite and fullerene–fullerite correlations);

fullerene intermediate family (2 < n < 3), including sp 2<n<3-hybridized (σ-sp 2<n<3 + (3 − n)π) carbons from numerous species of nanostructured fullerenes and hyperfullerenes, single-walled and multi-walled nanotubes, nanoscrolls, nanobarrels, nanocones and other nanostructures to macrostructural forms, such as fullerites, nanotubular crystals, hypothetical structures of haeckelites and schwarzites, as well as plenty of carbon/carbonaceous products, such as soot, carbon black, so-called “amorphous” carbon, pyrolitic carbon, coke, charcoal, activated carbon, molecular-sieve carbons, glassy carbon, carbide derived carbons, intermediate graphite-to-diamond transition phases, diamond-like carbon films and others, which are characterized by the intermediate between graphene and diamond hybridisation type (mainly, the fullerene family structures can be considered as distorted graphene related structures, where deviation from planarity is occured because of the partial heptagons and/or pentagons substitution for hexagons in graphene networks; special measures termed pyramidalization angle and/or curvature are used for evaluation of nonplanarity (deviation from graphene plane geometry) in fullerene family structures), which differ in the sp n-hybridisation (1 ≤ n ≤ 3) types of carbon structures.

The main structural characteristics, including crystal lattice parameters and densities, and relative thermal stabilities of the various carbon allotropic, polymorphic and polytypic forms, grouped under family headings within the overall hierarchy from submolecular to macrostructural scales, are summarized in Table 2.3.

Table 2.3 Characteristic and structural features of allotropic, polymorphic and polytypic carbon forms
Full size table

2.2 Thermal Properties

2H-graphite (or α-graphite) is the most stable thermodynamically form of carbon under standard conditions, so its standard enthalpy of formation (at 298.15 K) ΔH° f,298 is zero, similar to those characteristics for all other chemical elements. The generalized phase and transition diagram of carbon based on several main sources [28, 176–190, 535–538] is given in Fig. 2.1. In the region of moderate pressures, this variant of carbon phase diagram assumes the sequence of transformations with temperature (energy) increase from sp 3 (diamond) through sp 2 (graphite/graphene) to sp 1 (carbyte/carbyne), which is corresponding to the specific bond energy approach established for the carbon structures classification previously (see Sect. 2.1). The general thermodynamic properties of carbon (graphite) are summarized in Table 2.4.

Fig. 2.1
figure 1

The generalized phase and transformation diagram of carbon according to the several main sources [28, 176–190, 535–538]

Full size image
Table 2.4 General thermodynamic properties of graphite
Full size table

For the molar heat capacity c p = f(T, K), J mol−1 K−1, the following relationship for the range of temperatures from 298 to 2500 K is recommended [227]:

$$ c_{p} = 17.17 + \left( {4.27 \times 10^{ - 3} } \right)T-\left( {8.79 \times 10^{5} } \right)T^{ - 2} . $$
(2.1)

The heat capacity of carbon (graphite) materials increases rapidly with temperature up to 1000 °C, where it levels off at approximately 2 kJ kg−1 K−1 with the subsequent increase beginning at 2800−3000 °C (Fig. 2.2). The important remark, which should be made in regard to the thermal behaviour of carbon at ultra-high temperatures, is connected with its vaporization (especially, in vacuum). At moderate pressures carbon (graphite) does not melt but sublimates very intensively with vaporization rate exceeding 0.01 kg m−2 s−1 when temperature reaches more than 3000 °C. The values of standard molar entropy S°298, molar c p and specific c heat capacities, enthalpies (heats) of melting and vaporization, molar and specific enthalpy differences H T – H 298, vapour pressures and mass/linear vaporization rates are given in Addendum in comparison with other ultra-high temperature elements (refractory metals) in the wide ranges of temperatures.

Fig. 2.2
figure 2

Relative variations of some physical properties of common industrial graphite materials with temperature: c—heat capacity, α—thermal expansion, σ—mechanical strength, E—Young’s modulus, ρ—electrical resistivity and λ—thermal conductivity (based on several sources [41–43, 136, 138, 194, 209, 223–224])

Full size image

All vector-defined thermal properties of carbon (graphite) reflect extremely high anisotropy inherent to graphite crystals. The properties can vary considerably depending on geometrical directions in carbon (graphite) structures. The main anisotropy coefficient

$$ \zeta = {{a_{\text{para}} } \mathord{\left/ {\vphantom {{a_{\text{para}} } {a_{\text{perp}} }}} \right. \kern-0pt} {a_{\text{perp}} }}, $$
(2.2)

where a para, a perp are, respectively, the values of a physical property in parallel and perpendicular directions to the graphene basal plane (or axis of symmetry), for thermal conductivity of quasi-single crystalline graphite material, such as highly oriented pyrolytic graphite, reaches up to 200.

Around the hierarchy of graphene/graphite structures from nano- to macro- scales all the materials are characterized by highly anisotropic properties; e.g., carbon nanotubes are very good thermal conductors along the tube, but good insulators laterally to the tube axis (Table 2.5). Some kind of similar particularities are also common for the variety of carbon (graphite) products, including the graphite containing bulk materials, which are very different in structure and composition. The variation of thermal conductivity in the different directions of graphite crystals with temperature is shown in Fig. 2.3.

Table 2.5 Some physical properties of graphitic nanostructured carbons and quasi-single crystalline graphite at room temperature
Full size table
Fig. 2.3
figure 3

Variation of the thermal conductivity in the different directions of graphite crystals with temperature [41–43, 138, 209]

Full size image

For bulk carbon (graphite) materials produced in industry, thermal conductivity depends strongly on the various defects of their structures. The values of thermal heat capacity and thermal conductivity of various carbon products: industrial graphitized carbon materials, pyrolitic (pyrographite) and vitreous (glass-like) carbons, thermally expanded (exfoliated) graphite materials and carbon (graphite) filaments in commercial carbon fibres are given in Tables 2.6, 2.7, 2.8, 2.9, 2.10.

Table 2.6 Physical propertiesa of industrial graphitized carbon materials [11, 42, 136, 138, 192–194, 209–211, 215, 222–225, 228, 230, 237, 539]
Full size table

The thermal expansion–contraction behaviour of carbon (graphite) is very complicated (Fig. 2.4). To take into account this aspect is a question of vital importance in the materials design for ultra-high temperature applications. The magnitudes of linear thermal expansion coefficients for various industrial carbon (graphite) products are also presented in Tables 2.6–2.10.

Fig. 2.4
figure 4

Variations of linear thermal expansion coefficients in the different directions of graphite crystals with temperature [41–42, 209]

Full size image

For averaged highly oriented graphite and quasi-isotropic carbon (graphite), compared with other ultra-high temperature elements (refractory metals), the values of thermal conductivity and thermal expansion in the wide range of temperatures are summarized in Addendum.

2.3 Electro-Magnetic and Optical Properties

Together with thermal properties, electrical resistivity of carbon (graphite) materials and its property-temperature relationship are also extremely anisotropic and sensitive to the order/disorder of carbon atoms in graphitic materials structures. For well-ordered crystals (highly oriented pyrolytic graphite), the variations of specific electrical resistance in the general crystallographic directions and in the wide range of temperatures are presented in Fig. 2.5.

Fig. 2.5
figure 5

Variations of specific electrical resistance in the different directions of graphite crystals with temperature (Inset–for the parallel direction to the basal planes in temperature (T, K) logarithm scale) [41, 209]

Full size image

Indeed, the character of these plots differs essentially from the electrical resistivity—temperature relationships observed for the common graphite materials (see Fig. 2.2), which are far from the ideal structure. The values of electrical resistance of graphitic nanostructured carbons, various industrial graphitized carbon materials, pyrolitic (pyrographite) and vitreous (glass-like) carbons, thermally expanded (exfoliated) graphite materials and carbon (graphite) filaments in commercial carbon fibres are given in Tables 2.5–2.10.

Carbon (graphite) is highly diamagnetic with molar magnetic susceptibility χmol (SI) = −75.4 × 10−6 cm3 mol−1 in parallel direction to the basal graphene planes at room temperature [3, 539–540], the χmol versus temperature relationships for both main directions in the graphite crystals are shown in Fig. 2.6.

Fig. 2.6
figure 6

Variations of molar magnetic susceptibility in the different directions of graphite crystals with temperature [539]

Full size image

The general optical properties of carbon (graphite) for wavelength λ = 0.620 μm are following: index of refraction (single crystal)—2.6, index of absorptance (single crystal)—1.4, reflective index under normal incidence—0.24 (polished graphite) and 0.35 (cleaned single crystal surface) [6]. In visible-light spectrum the monochromatic emittance (spectral emissivity) ελ of graphite is near to that of a grey body and varies for λ = 0.65 μm from 0.77 (well-polished surfaces) to 0.95 (roughened surfaces). The integral emittance εT of graphite ranges from 0.6 to 0.9. The temperature variations for the both coefficients of emittance are linear:

$$ \varepsilon = \varepsilon_{0} \pm aT, $$
(2.3)

with the positive values of a for integral emittance εT and negative value—for spectral emittance ελ (λ = 0.665) [138]. The recommended values of electrical resistivity, magnetic susceptibility, integral and spectral emittances and thermoionic emission characteristics for carbon (graphite) materials are given in comparison with other ultra-high temperature elements (refractory metals) in Addendum.

2.4 Physico-Mechanical Properties

Hardness of carbon (graphite) materials ranges widely depending on their macro- and microstructures and bulk densities. To compare this property for the various graphitic products or with other ultra-high temperature materials is often connected with certain difficulties because the hardness measurements have been made in practice by differing methods (Shore HS, Vickers HV, Rockwell HR, Brinell HB, Knoop HK, Mohs HM scales [192]). For the main types of carbon (graphite) materials approximate hardness values are following:

  • industrial structural and electrode graphitized (molded) products—HV 30−100 kgf mm−2 (0.3–1 GPa) [42], HM 1, HR 70−110 or HS 50−80 hardness numbers [192];

  • vitreous carbon (graphite)—HV 150–340 kgf mm−2 (1.5–3.4 GPa);

  • pyrolytic carbon (graphite)—HV 140–370 kgf mm−2 (1.4–3.7 GPa) [42, 136, 192].

At ambient temperatures, because of pores, flaws and microstructural defects concentrating the applied mechanical stresses, carbon (graphite) polycrystalline materials fails at much lower level than the theoretical strength of the graphite crystal, and mechanical failure occurs by absolutely brittle fracture. The flaws and defects are gradually annealed, stresses are relieved and plastic deformation becomes more probable with increasing temperature [42]. This evolution results in the general increase of the mechanical properties of graphite, so at the temperature range of about 2500 °C the strength of graphitized carbon materials is almost twice the value corresponding to room temperature. Theoretical predictions and experimental observations of tensile strength of graphitic nanostructured carbons are shown in Table 2.5. The values of strength characteristics of structural, electrode and nuclear graphite materials for the different types of mechanical loading (tension, flexure and compression) are given in Table 2.6. Commonly, the physico-mechanical properties of graphitized materials, because of the preferential orientation of graphite grains (microcrystallites), vary with parallel and perpendicular directions to the axis of processing (extrusion, molding, pressing etc.). However, in spite of the fact that the individual crystals of graphite always perform super-high grade of anisotropy, special technological routes are elaborated for the industrial production of quasi-isotropic graphite materials [11]. The ratios between tensile σt, flexural (bending) σf and compressive σc strength characteristics for structural graphite materials

$$ \sigma_{f} /\sigma_{t} \approx 2 $$
(2.4)

ranges from 1.5 to 2.1, and

$$ \sigma_{c} /\sigma_{f} \approx 2 $$
(2.5)

ranges from 1.6 to 2.9 [194]. The various strength properties of pyrolytic carbon (graphite) materials are presented in Table 2.7. The flexural (bending) and compressive strengths of vitreous (glass-like) carbon materials are given in Table 2.8 and tensile strength of thermally expanded (exfoliated) graphite materials is included in Table 2.9. The tensile strength and strain to failure (elongation) of carbon (graphite) fibres are given in Table 2.10. Data on the fracture toughness (critical stress intensity factor) of some nuclear graphites are included in Table 2.6.

Table 2.7 General physical properties of pyrolytic carbon (graphite) materialsa
Full size table
Table 2.8 General physical propertiesa of common vitreous (glass-like) carbon materials [11, 42, 136, 207, 209, 539]
Full size table
Table 2.9 General physical propertiesa of common thermally expanded (exfoliated) graphite materials [212–213, 216]
Full size table
Table 2.10 General physical propertiesa of carbon (graphite) filaments in commercial carbon fibres [84, 136, 207, 217–221]
Full size table

Although the strength of graphite rises with temperature increase, at the same time creep resistance of graphitized materials falls. Commonly, the creep characteristics of carbon (graphite) materials are higher in parallel direction to the preferential orientation of the normals to the basal planes of graphite grains (or in parallel direction to molding axis) than analogous characteristics in perpendicular directions. The strain rate of graphitized materials during steady-state stage of creep is defined by general relationship

$$ \dot{\varepsilon } = C\left( {\frac{\sigma }{{\sigma_{b} }}} \right)^{n} \exp \left( { - \frac{Q}{RT}} \right) $$
(2.6)

where \( \dot{\varepsilon } = \partial \varepsilon /\partial t = \partial l/l\partial t \) is the creep rate in s−1, C ≈ 40 is the constant independent of applied stress σ and temperature T in K, σ b ultimate strength, n ≈ 4 is the creep exponent constant, Q ≈ 210 kJ mol−1 is the activation energy of creep and R is the gas constant [194, 908]. At the same temperature the creep rate of graphitized materials is higher in vacuum and reduces with inert gas pressure increase.

Quasi-single crystalline graphite (e.g. highly oriented pyrolytic graphite) is a typical transversely isotropic material, and most graphitic and graphite containing materials applied in practice possess the infinite order axis of symmetry as well. In the expression for the elastic constants of graphite crystals

$$ \Upsigma_{i} = c_{ij} \Upxi_{j} \left( {i,j = \, 1, \, 2, \ldots , \, 6} \right), $$
(2.7)

linking the stresses Σ i and the strains Ξ j in the low strain limit, only 12 parameters (stiffness coefficients or modules) c ij are non-zero: c 11 = c 22, c 12 = c 21, c 13 = c 31 = c 23 = c 32, c 33, c 44 = c 55 and c 66 = (c 11–c 12)/2.

Thus, the elasticity tensor for single crystal graphite material is symmetrical, and the stress–strain relationship (representation of Hooke’s law) expressed by the matrix notations is read [217, 229, 231]:

$$ \left\{ {\begin{array}{*{20}c} {\sigma_{xx} } \\ {\sigma_{yy} } \\ {\sigma_{zz} } \\ {\sigma_{yz} } \\ {\sigma_{xz} } \\ {\sigma_{xy} } \\ \end{array} } \right\} = \left( {\begin{array}{*{20}c} {c_{11} } & {c_{12} } & {c_{13} } & 0 & 0 & 0 \\ {c_{12} } & {c_{11} } & {c_{13} } & 0 & 0 & 0 \\ {c_{13} } & {c_{13} } & {c_{33} } & 0 & 0 & 0 \\ 0 & 0 & 0 & {c_{44} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {c_{44} } & 0 \\ 0 & 0 & 0 & 0 & 0 & {\frac{1}{2}(c_{11} - c_{12} )} \\ \end{array} } \right)\left\{ {\begin{array}{*{20}c} {\varepsilon_{xx} } \\ {\varepsilon_{yy} } \\ {\varepsilon_{zz} } \\ {\varepsilon_{yz} } \\ {\varepsilon_{xz} } \\ {\varepsilon_{xy} } \\ \end{array} } \right\} $$
(2.8)

or in the inverse form with compliances s ij :

$$ \left\{ {\begin{array}{*{20}c} {\varepsilon_{xx} } \\ {\varepsilon_{yy} } \\ {\varepsilon_{zz} } \\ {\varepsilon_{yz} } \\ {\varepsilon_{xz} } \\ {\varepsilon_{xy} } \\ \end{array} } \right\} = \left( {\begin{array}{*{20}c} {s_{11} } & {s_{12} } & {s_{13} } & 0 & 0 & 0 \\ {s_{12} } & {s_{11} } & {s_{13} } & 0 & 0 & 0 \\ {s_{13} } & {s_{13} } & {s_{33} } & 0 & 0 & 0 \\ 0 & 0 & 0 & {s_{44} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {s_{44} } & 0 \\ 0 & 0 & 0 & 0 & 0 & {2(s_{11} - s_{12} )} \\ \end{array} } \right)\left\{ {\begin{array}{*{20}c} {\sigma_{xx} } \\ {\sigma_{yy} } \\ {\sigma_{zz} } \\ {\sigma_{yz} } \\ {\sigma_{xz} } \\ {\sigma_{xy} } \\ \end{array} } \right\}. $$
(2.9)

The compliance coefficients are readily expressed in terms of the stiffness coefficients:

$$ 2s_{11} = \frac{{c_{33} }}{{c^{2} }} + \frac{1}{{c_{11} - c_{12} }}, $$
(2.10)
$$ 2s_{12} = \frac{{c_{33} }}{{c^{2} }} - \frac{1}{{c_{11} - c_{12} }}, $$
(2.11)
$$ s_{13} = - \frac{{c_{13} }}{{c^{2} }}, $$
(2.12)
$$ s_{33} = \frac{{c_{11} + c_{12} }}{{c^{2} }}, $$
(2.13)
$$ s_{44} = \frac{1}{{c_{44} }}, $$
(2.14)
$$ c^{2} \equiv c_{33} (c_{11} + c_{12} ) - 2c_{13}^{2} = \frac{1}{{s^{2} }} > 0, $$
(2.15)
$$ s^{2} \equiv s_{33} (s_{11} + s_{12} ) - 2s_{13}^{2} . $$
(2.16)

The reciprocal expressions of stiffness coefficients in terms of compliance coefficients have the same form if c 2 is replaced with s 2.

In particular, there are seven elastic characteristics of graphitic materials, which can be defined and measured (various designations of them applied in the literature are given in brackets) [217, 226, 229, 231]:

E 1 (E, E x , E a or E 11)—in-plane Young’s modulus (for directions within the isotropic (isometric) plane, or in the parallel directions to the basal graphene planes; for carbon (graphite) high-modulus fiber filament—longitudinal modulus, commonly);

E 2 (E′, E z , E c or E 33)—out-of-plane Young’s modulus (for the direction perpendicular to the isotropic (isometric) plane, or in the perpendicular direction to the basal graphene planes; for carbon (graphite) high-modulus fiber filament—transversal modulus, commonly);

G 12 (G or G xy )—Coulomb’s (shear) modulus for the isotropic (isometric) plane, or in the parallel planes to the basal graphene planes;

G 13 (G′ or G yz = G zx ,)—Coulomb’s (shear) modulus for planes normal to the isotropic (isometric) plane, or in the perpendicular planes to the basal graphene planes;

ν12 (ν or ν xy )—Poisson’s ratio (major), which characterizes the contraction within the isotropic (isometric) plane due to forces applied within this plane;

ν13 (ν′ or ν xz )—Poisson’s ratio, which characterizes the contraction within the isotropic (isometric) plane due to forces applied in the direction perpendicular to it;

ν23 (ν′′ or ν zx )—Poisson’s ratio, which characterizes the contraction in the direction perpendicular to the isotropic (isometric) plane due to forces applied within this plane.

However, only five from seven characteristics mentioned above are independent.

The equality E 1ν13 = E 2ν23 exists because of the symmetry of the elasticity tensor.

Young’s and Coulomb’s (shear) moduli and Poisson’s ratios are associated with the stiffness coefficients c ij in the following forms:

$$ \begin{gathered} \left( {\begin{array}{*{20}c} {1/E_{1} } & { - \nu_{12} /E_{1} } & { - \nu_{13} /E_{2} } & 0 & 0 & 0 \\ { - \nu_{12} /E_{1} } & {1/E_{1} } & { - \nu_{13} /E_{2} } & 0 & 0 & 0 \\ { - \nu_{13} /E_{2} } & { - \nu_{13} /E_{2} } & {1/E_{2} } & 0 & 0 & 0 \\ 0 & 0 & 0 & {1/G_{13} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {1/G_{13} } & 0 \\ 0 & 0 & 0 & 0 & 0 & {1/G_{12} } \\ \end{array} } \right) \hfill \\ = \left( {\begin{array}{*{20}c} {c_{11} } & {c_{12} } & {c_{13} } & 0 & 0 & 0 \\ {c_{12} } & {c_{11} } & {c_{13} } & 0 & 0 & 0 \\ {c_{13} } & {c_{13} } & {c_{33} } & 0 & 0 & 0 \\ 0 & 0 & 0 & {c_{44} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {c_{44} } & 0 \\ 0 & 0 & 0 & 0 & 0 & {\frac{1}{2}(c_{11} - c_{12} )} \\ \end{array} } \right)^{ - 1} \hfill \\ \end{gathered} $$
(2.17)

The experimental estimates of the elastic properties of 2H-graphite crystals were based on rather indirect data on the ensembles of small pyrolytic graphite crystals with close orientations along the hexagonal axis of symmetry (axis c) and with a wide orientation scatter in perpendicular directions [232, 234]; later more direct experimental data on 2H-graphite single crystals were extracted using the method of inelastic x-ray scattering [233]. These experimental data together with some theoretical estimates for 2H- and 3R-graphites are shown in Table 2.11.

Table 2.11 Elastic properties of graphite crystals at room temperature
Full size table

The condition for the isotropic character of media is

$$ 2{{c_{44} } \mathord{\left/ {\vphantom {{c_{44} } {\left( {c_{11} -c_{12} } \right)}}} \right. \kern-0pt} {\left( {c_{11} -c_{12} } \right)}} = 1, $$
(2.18)

so substituting the values for these elastic constants of graphite, this ratio becomes 0.0004 [136], which means that the Young’s modulus varies appreciably with orientation and graphite materials are extremely anisotropic. The Young’s modulus of graphite crystals as a function of angle θ with c axis (perpendicular to the basal graphene planes) is symmetric for 2H-graphite (or near-symmetric for 3R-graphite) reaching in both cases the highest values (E > 1 TPa) with sharp maximum at θ = π/2 and the lowest values (E < 5 GPa) with low-grade minimum at θ ≈ 0.2π. For the analogous angular dependence of Poisson’s ratio ν xy , the asymmetry about θ = π/2 for 3R-graphite is somewhat larger; ν xy is limited by the value 0.16 reached in the neighborhood of θ = π/2 and is rapidly reduced for both graphite polytypes. In the range of angles θ = (0.13−0.38)π Poisson’s ratio ν xy becomes negligible (less than 0.01) and even negative. In the transverse direction, orthogonal to the previous one, the behaviour of Poisson’s ratio ν zx is quite different, since its value can be higher than the upper limit of Poisson’s ratio for a elastic isotropic medium (its thermodynamic limits are −1 < ν < 0.5) [229]. The values of Young’s modules for graphitic nanostructured carbon forms are summarized in Table 2.5. For large stresses in reality, the stress-strain relationship for polycrystalline graphite materials is non-linear and approximated by the equation of Jenkins [237]

$$ \varepsilon = A\sigma + B\sigma^{2} , $$
(2.19)

where A is equal to s 33 for specimens cut off in parallel direction to the preferential orientation of the normals to the basal planes of graphite flake-like grains (against-grain specimens) or s 11 for specimens cut off in perpendicular direction to the preferential orientation of the same normals (with-grain specimens), and in accordance to Jenkins’ model

$$ B = {{2\varepsilon_{0} } \mathord{\left/ {\vphantom {{2\varepsilon_{0} } {\sigma_{\text{m}}^{2} }}} \right. \kern-0pt} {\sigma_{\text{m}}^{2} }}, $$
(2.20)

where ε0 is the longitudinal residual strain after the graphite has been subjected to a maximum stress σm. For various industrial graphitized carbon products (for specimens cut off in different directions), the values of A range from 0.044 to 0.207 GPa−1 and B range from 0.232 to 3.262 GPa−2 [237]. The elasticity of commercially available graphitic products is considerably different from those values of the ideal crystals since they are controlled by the preferential crystallite orientation, porosity and structural defects. The elastic properties of various industrial graphitized carbon materials, pyrolitic (pyrographite) and vitreous (glass-like) carbons and carbon (graphite) fibres are given in Tables 2.6–2.8 and 2.10. The temperature variation of Young’s modulus and strength characteristics of averaged industrial graphitized materials is presented in Fig. 2.2.

For averaged highly oriented graphite and quasi-isotropic carbon (graphite), compared with other ultra-high temperature elements (refractory metals), the values of physico-mechanical (strength, elasticity) properties in the wide range of temperatures are summarized in Addendum.

2.5 Nuclear Physical Properties

The isotopes of carbon (standard atomic mass—12.0107 u) from 8C to 22C and their general characteristics are summarized in Table 2.12; the naturally occurring isotopes are listed in order of decreasing abundance, and unstable radioactive isotopes—in order of decreasing half-life period of decay.

Table 2.12 General characteristics of the isotopes of carbon [3, 6, 576–578]
Full size table

Nuclear physical properties of carbon (isotopic mass range, total number of isotopes, thermal neutron macroscopic cross sections, moderating ability and capture resonance integral), compared with other ultra-high temperature elements (refractory metals), are given in Addendum.

2.6 Chemical Properties

The specific electron configuration of carbon atoms has been proved in the numberless variety of carbon containing compounds formed by the chemical interaction between carbon and other elements of the periodic table. Organic chemistry as a special discipline is devoted directly to the carbon compounds with hydrogen, oxygen and nitrogen and their multi-component derivatives, which are termed organic compounds and, correspondingly, differed from inorganic compounds. The various inorganic compounds of carbon are classified in the following types:

  • simple molecular compounds (with p-elements of groups 15–17: halogens (F, Cl, Br, I), chalcogens (O, S, Se) and pnictogens (N, P)), covalent bonded;

  • salt-like carbides (or acetylenides, compounds with s-elements of groups 1–2: alkali metals (Li, Na, K, Rb, Cs) and alkaline earth metals (Be, Mg, Ca, Sr, Ba) and d-elements of groups 11–12: Cu, Ag, Au, Zn, Cd, Hg), mainly ionic or covalent-ionic (e.g. Be and Mg carbides) bonded;

  • interstitial carbides (or metal-like carbides, compounds with d-elements of groups 4–6: Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, W), mainly metallic bonded;

  • intermediate carbides (compounds with d- and f-elements of group 3: Sc, Y, lanthanoides (La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu) and actinoids (Th, Pa, U, Np, Pu, Am) and light d-elements of groups 7–10: Mn, Fe, Co, Ni), ionic-metallic or covalent-metallic bonding;

  • covalent carbides (compounds with p-elements of groups 13–14: B, Al, Si), mainly covalent or covalent-ionic (e.g. Al carbides) bonded;

  • graphitic/graphene and fullerene compounds (or intercalation (lamellar) compounds, with some elements, ions and molecules), mainly molecular bonded;

  • carbon containing complex compounds (ternary, quaternary and higher order compounds) with the combination of the different types of chemical bonding.

The comprehensive information on the chemical interaction of carbon with all the elements of the periodic table is given in Table 2.13. The carbon containing systems and corresponding binary compounds are considered there in accordance to the groups of elements from 1 to 17.

Table 2.13 Chemical interaction of carbon (graphite) with elements of the periodic table (binary systems in accordance to the groups of elements)a
Full size table

The data on the selected ternary, quaternary, quasi-binary, quasi-ternary and multi-component carbon containing systems, which are the most important for the design, manufacture and application of ultra-high temperature materials, are summarized in Table 2.14. The composition and temperature stability regions for the main binary and ternary carbon containing high temperature phases are given in Tables 2.13, 2.14 taking into account the spread of numerical magnitudes available in literature currently.

Table 2.14 Chemical interaction of carbon (graphite) with elements and compounds at elevated, high and ultra-high temperatures (selected ternary, quaternary, quasi-binary, quasi-ternary and multi-component systems in alphabetical order)a
Full size table

The results obtained by Wang et al. [646] indicate that at room temperature surface energy of graphene, graphene oxide and natural graphite flakes are 47, 62 and 55 mJ m−2, respectively. Before, a 150 mJ m−2 value for the surface energy of the basal plane of graphite was measured at room temperature by wetting experiments [910]. The formation of a surface-orientated perpendicular to the basal plane needs the rupture of C–C chemical bonds. The surface energy for this orientation has not been determined experimentally, but is expected to be an order of magnitude higher than that of the basal plane. In the case of vitreous carbon, the value was measured and found to be 32 mJ m−2 [910]. This is five times lower than that of the graphite basal plane and may be explained by the low density of vitreous carbon. The average static contact angles of graphene/graphite with some liquids, which were measured by the droplet on the film surface at room temperature, are listed in Table 2.15 [646, 909]. The quantitative characteristics of the wettability of graphite materials by some non-ferrous metal alloys (melts) at elevated and high temperatures are shown in Table 2.16 [530, 661, 910], and wetting characteristics in alkali metal halide–graphite systems are given in Table 2.17 [910].

Table 2.15 Average static contact angles (in degrees) of graphite, graphene oxide and graphene with some liquids measured by the droplet on the film surface at room temperature [646]a
Full size table
Table 2.16 Wettability of graphite materials by some non-ferrous metal alloys (melts) in vacuum [530, 661, 910]
Full size table
Table 2.17 Wetting contact angles (in degrees) of pure molten halides MeX (Me = Li, Na, K, Rb; X = F, Cl, Br) on graphite in dry inert gas atmosphere at 1000 °C [910]
Full size table

The character of chemical interaction and general reactions of carbon (graphite) with common chemicals (solids, aqueous solutions) and complex gases are summarized in Table 2.18.

Table 2.18 The interaction of carbon (graphite) with some chemicals and complex gases [5, 82, 238, 531–532]
Full size table

The self-diffusion characteristics of carbon atoms, diffusion characteristics in the carbon—element and carbon—chemical compound systems in the wide range of temperatures, and summarized data on the physico-chemical interaction of carbon with the elements of periodic table are given in Addendum.