ISSN 1995-0780, Nanotechnologies in Russia, 2015, Vol. 10, Nos. 1–2, pp. 18–24. © Pleiades Publishing, Ltd., 2015. Original Russian Text © S.A. Gubin, I.V. Maklashova, E.I. Dzhelilova, 2015, published in Rossiiskie Nanotekhnologii, 2015, Vol. 10, Nos. 1–2.
On the Effect of Size, Shape, and Internal Structure on Phase Equilibrium in Graphite and Diamond Nanocrystallites S. A. Gubin, I. V. Maklashova, and E. I. Dzhelilova National Research Nuclear University, Moscow, Russia e-mail:
[email protected] Received July 16, 2014; accepted for publication October 23, 2014
Abstract—The properties of nanodispersive carbon are different from those in its typical bulk-crystalline state. For nanographite and nanodiamond, the change in thermodynamic potentials caused by the nanodispersive state depends on the size, shape and arrangement of carbon atoms in a nanoparticle. In this work we developed a technique allowing us to estimate the enthalpy of formation and entropy in nanographite and nanodiamond as functions of size, shape, and internal structure of nanocrystals. The relevant nanographite– nanodiamond phase diagrams have been calculated. DOI: 10.1134/S1995078015010073
increases with a decreasing size of spherical nanoparticles. The reverse statement of increasing graphitization temperature with a decreasing size of spherical carbon nanoparticles [14] seems doubtful, since it contradicts the experimental data [15–19], which clearly indicate a decrease in graphitization temperature with an increase in particle size. One can thereby conclude that the phase transition of spherical nanodiamond particles into graphite occurs at higher pressures and lower temperatures than the phase transition of typical bulk-crystalline diamond. It is possible to estimate the change in thermodynamic potentials of nanodispersive substance by using thermodynamic relations. In this case the sizes of carbon nanoparticles are usually assumed to be not too small to considerably modify the internal structure of a substance at a given pressure, temperature, and work of formation of the unit nanoparticle surface. This approach is not valid for evaluating the properties of small clusters, but it is valid for calculating the properties of nanoparticles that have on the order of 103–10 4 and more atoms in crystal. In this work the proper EoS for graphite and diamond nanoparticles have been written on basis of EoS obtained via the technique [20] in [21] in the Mie– Gruneisen form for the relevant phases of bulk-crystallite carbon. Table 1 shows respectively the coefficients of the Mie–Gruneisen EoS for the bulk-crystallite carbon. As is assumed, the structure, the interatomic distances, and the binding energies are equal for macroand nanodispersive materials. The only difference is
INTRODUCTION Carbon nanoparticles with various phase states and intricate shapes were discovered in the latter half of the 20th century. Among them, detonation diamond nanoparticles [1], carbon nanotubes, and onionlike nanoparticles formed of several layers of graphite or diamond films, as well as fullerenes and carbines, were obtained. The combination of unique physicochemical properties makes carbon nanoparticles promising objects not only for understanding the processes under combustions and explosion, but also for their industrial applications. In spite of numerous works, the phase stability regions and synthesis conditions of carbon nanoparticles are not adequately explored. If the phase diagram of carbon in typical bulk-crystalline graphite and diamond phases has been studied for a long time [2–8] and is well known even in the regions of high pressures and temperatures, only a few studies dedicated to investigating phase transformations and thermodynamic stability regions of nanodispersive carbon phases are reported in the literature. There are almost no valid equations of state (EoS) which allow estimating the thermodynamic properties of diamond and graphite nanoparticles in the wide region of pressure and temperature changes depending on the size, shape, and internal structure of nanoparticles. The earliest phase diagrams of nanocarbon [1, 9, 10] are nowadays confirmed both experimentally and numerically. As follows from the MD calculations [11–13], at constant pressure the nanodiamond-to-nanographite transition point decreases and the transition pressure 18
ON THE EFFECT OF SIZE, SHAPE, AND INTERNAL STRUCTURE ON PHASE Mie–Gruneisen EoS coefficients for diamond and graphite Diamond
Graphite
1/v0, g/cm3
3.515
2.25
θ0, K
1320
1090
a0
5.4
–3.74
q0
0.0
0.0
ax
7.3
1.5
C1, GPa
–36.2032
∫
27234.7
C3, kJ/g
–869.818
–9082.33
Em, kJ/g
0.15826
0.0
N M, g/mol as
1 12.011 0.0
1 12.011 0.0
that the nanoparticle exhibits high surface energy due to its small size and thus induces the relevant changes in thermodynamic potentials (enthalpy and entropy) in the substance. In other words, for nanoparticles, the caloric EoS is changed, but the thermal one is equal to that for typical bulk-crystalline carbon phases. To calculate the thermodynamic states of different phases in the condensed nanocarbon in view of dispersion (size), shape, and arrangement of carbon atoms in a nanoparticle, the changes in enthalpy and entropy of carbon nanocrystals must correctly be taken into account. The correction of thermodynamic potentials allows estimating the enthalpy of formation and entropy values in carbon nanoparticles under standard conditions depending on size, shape, and internal structure of carbon nanoparticles. The change in thermodynamic potentials has been estimated in view of the work of a nanoparticle surface formation. In the general case for the Gibbs energy of the condensed matter, the basic equation of thermodynamics is valid in terms of the work of surface formation [22]:
dG = −SdT + VdT + σ dA + μ dN . where G is the Gibbs energy, S is the entropy, T is the temperature, р is the pressure, V is the volume, μ is the chemical potential, N is the number of moles of substance, A is the surface area, and σ is the work of formation of the unit surface area (the surface energy):
( )
(1) σ = ∂G . ∂ A T , p,N As follows from relation (1), at constant temperature and pressure, the change in the surface area for the definite amount of condensed matter is accompanied by the change in the Gibbs energy:
dGT , p = σ dA. Vol. 10
( )
(2)
( ) dA .
(3)
Δ ST , p = − ∂σ ∂T
∫
–1307.73
543.5759
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and the changes in enthalpy and entropy of substance at the finite change of the surface area under the above conditions are expressed as the equations
⎡ ⎤ Δ H T , p = ⎢σ − T ∂σ ⎥dA , ∂T p ⎦ ⎣ A
C2, GPa
Nos. 1–2
19
A
p
The specific surface energy of many substances is known to be the linear function of temperature almost to its critical value [23]. We thus assume that it is also valid for the considered carbon nanoparticles; i.e., the following condition is held: (4) σ = σ 0 − α(T − T0 ) , where σ0 is the work of formation of the unit surface at temperature Т0, usually being 20˚С, and α = const is the temperature coefficient. For most single-component substances, α ≈ 0.1 mJ/(m2 K) [23]. The changes in enthalpy and entropy of the substance caused by its nanodispersive state are thus due to the change in the surface area and depend on the temperature. Consequently, the whole influence of high surface energy of nanoparticles on the thermodynamic (caloric) functions of a nanodispersed matter leads to the changes in enthalpy and entropy of this substance as the functions of temperature and can be taken into account by the correction of the enthalpy of formation and entropy of matter at standard conditions. The changes in entropy and enthalpy of formation of a nanodispersive substance with temperature were brought to the calculation of these magnitudes (the enthalpy of formation of substance and entropy) for nanodispersive substances at standard conditions from the relations (2)–(4) without changes in thermal EoS for the definite carbon phases. The considered approach neglects the modifications in the internal structure of the nanomaterial. The structure and the surface of crystals are assumed to be ideal. The gas adsorption, as well as the formation of oxide films at the nanoparticle surface, was not taken into account. For the concrete nanodispersive material, the ° ) change in standard enthalpy of formation Δ(Δ f H 298 depends on the sizes, shapes, and internal structure of the atom arrangement in the nanoparticle; i.e., these parameters affect the value of the nanoparticle surface area [9]. Therefore, when calculating the thermodynamic and thermophysical properties and phase diagram of nanodispersive material, the chemical potentials of the nanodispersive condensed phases are calculated on the basis of the above thermal EoS for the typical bulk-crystallite carbon phases in terms of the corrections in enthalpy and entropy of the substance due to 2015
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1
109ο28'
2
109ο28'
3
Fig. 1. Crystalline structure of carbon phases: (1) hexagonal diamond (lonsdalite), (2) cubic diamond, and (3) graphite.
its nanodispersive state, shape, and internal structure of nanocrystals. The phase equilibrium conditions for nanodispersive carbon phases are mostly defined by the shape and size of nanoparticles. One can thus define several shapes of graphite and diamond nanoparticles: planar films, plane-oriented flakes and sheets, bulk-oriented prisms (fibers), and compact nanocrystals (the minimum distance between the prism vertexes and the center for the definite amount of atoms), as well as according to the thermodynamically advantageous shape (the minimum value of the surface energy). Let us choose for consideration two structures of diamond nanoparticles and one structure of graphite particles. For diamond particles, such structures are an octahedron (cubic diamond) and a regular prism with a hexagon in its base (lonsdalite), while for graphite structures there is only a regular prism with a hexagon in its base. The internal structures of the carbon atom arrangement in graphite are presented in the form of positioned nearby and one after another prisms with a hexagon in the base. The unification of such prisms positioned nearby forms two layers of graphite nets
separated by the prism height, which is equal to the bond length in graphite. The diamond structures are constructed by the octahedral (cubic diamond) and regular prisms with a hexagon in the base (lonsdalite) (Fig. 1). The nanodispersive graphite and diamond structures of different shapes can therefore be constructed of prisms and octahedra, like the brick building. Knowing the amount and positions of such bricks in the nanoparticle with a definite shape and size, one can calculate the amount of surface carbon atoms and estimate the surface energy of the nanoparticle. The work of formation of the unit nanoparticle surface can be determined via the number of dangling bonds and the area of its surface:
N p bind , (5) σ 0 = 1 e bind 2 A where еbind is the energy of a single bond, Nр bind is the number of dangling bonds at the nanoparticle surface, and А is the surface area of a nanoparticle. The value σ0 is defined by the ratio of the dangling bond energy to the area of the surface forming at the same time, where the coefficient 1/2 takes into
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account А as the area of only one of the two forming surfaces. The octahedron is characterized by only one geometric size—the edge length С. The regular prism with a hexagon in the base is described by two geometrical characteristics—the base edge length B and the prism height D. The prism shape can be defined by eccentricity E = D as the similarity parameter. This allows 2B using the eccentricity for obtaining the approximate characteristics of nanodispersive carbon. The dependencies for other crystal shapes exhibit a more complex form with an arbitrary eccentricity value. Under the equilibrium of nanodispersive carbon phases, crystals of both phases with the same amount of atoms were considered. There is still the open question of the ratio of different nanocristalline phases under equilibrium; i.e., what are the shapes and sizes of nanodiamond and nanographite crystals? The surface energy of the nanodispersive substance has been determined via work that has to be implemented for opening the surface atom bonds
tion on the number of atoms in nanocrystals and eccentricity value E brings the following expressions: for nanodiamond in the form of a regular hexagonal prism:
(6) L = 1 e bind KN p bind , 2 where L is the work, еbind is the energy of a single bond, К is the amount of crystals of a substance, Nр bind is the number of dangling bonds at the surface of a crystal, and the coefficient 1/2 takes into consideration the energy that has to be consumed for the formation of two dangling bonds equal to the energy of single bond. The change in standard enthalpy of formation can be expressed through the work of formation of a surface:
(10)
N ° ) = L = 1 E bind p bind , Δ(Δ f H 198 N 2 N at
(7)
° is the standard enthalpy of formation of where Δ f H 198 ° = 0), N = K N at is the substance (for graphite Δ f H 198 NA amount of moles of the substance, Nat is the number of atoms in a single crystal, NА is the Avogadro’s number, and Ebind = ebindNA is the binding energy per mole. The direct calculations of the atoms in crystal, of the number of dangling bonds at the nanoparticle surface, and of the geometric characteristics of crystal bring the expressions for the characteristic size (the octahedron edge length С or the prism base side length В) and the molar area of a crystal surface АМ depending on the amount of atoms in crystal. The above expressions enabled us to obtain the abundant values of standard formation enthalpy as the functions of the amount of carbon atoms in nanocrystals. The approximation of denotations of the changed abundant values for the standard enthalpy of formaNANOTECHNOLOGIES IN RUSSIA
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° ) = ⎛ 481863 + 19289 ⎞ ⎛ E ⎞ Δ(Δ f H 198 ⎜ ⎟ E ⎠ ⎜⎝ N at ⎟⎠ ⎝ for nanographite:
(J/mol), (8)
1/3
° ) = ⎛ 552500 + 22751⎞ ⎛ E ⎞ (J/mol). (9) Δ(Δ f H 198 ⎜ ⎟ E ⎠ ⎜⎝ N at ⎟⎠ ⎝ The above dependencies are obtained for the atom amount in a crystal from 10 4 to 107 and over an eccentricity range from 1 to 1000. For the considered shapes of graphite and diamond crystals, we express the changes in the abundant ° ) and enthalpy and entropy of nanoparticles Δ(Δ f H 198 ΔS as follows: (i) for the compact shape of hexagonal diamond crystals: ° ) = 670.5N at−1/3 (kJ/mol), Δ(Δ f H 198
− 1/3 Δ S = 9.6N at (J/(mol K)); (ii) for the compact shape of nanographite crystals:
° ) = 526.6N at Δ(Δ f H 198
− 1/3
(kJ/mol),
(11)
− 1/3
Δ S = 12.5N at (J/(mol K)); (iii) for thermodynamically favorable shape of cubic diamond nanocrystals: ° ) = 586.5N at Δ(Δ f H 198
− 1/3
(kJ/mol),
(12)
− 1/3
Δ S = 8.9N at (J/(mol K)); (iv) for thermodynamically favorable shape of graphite nanocrystals:
° ) = 185.1N at Δ(Δ f H 198
− 1/3
(kJ/mol),
(13)
− 1/3 Δ S = 60.6N at (J/(mol K)); where Nat is the number of atoms in nanocrystal. We thus obtained the changes in standard enthalpy of formation and entropy at normal conditions as the functions of the number of atoms in crystal for two different types of shapes of nanographite and nanodiamond particles. Now we consider three characteristic cases. First, we analyze the equilibrium between thermodynamically favorable shapes of nanodiamond and nanographite, which are, respectively, the octahedron and the film. These forms correspond to the minimum over-enthalpy value for a definite number of atoms. For the graphitic prism, the minimum value of the
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Δ(ΔH), kcal/mol 20 18 16 14 12 1 3 5 2 10 8 6 4 2 0 0.3 3
4
30
B, nm
Fig. 2. Change in enthalpy of formation for diamond as the function of the characteristic size of crystal for various nanoparticle shapes: (1) fiber with eccentricity Е = 10, (2) sheet with eccentricity Е = 0.1, (3) compact shape of diamond with a hexagon in the base, (4) thermodynamically favorable diamond shape (cubic diamond), and (5) thermodynamically favorable hexagonal diamond shape (lines 3 and 5 are coincident).
Δ(ΔH), kcal/mol 20 18 16 14 1 12 10 8 6 3 4 2 2 4 0 104 105 102 103
106
107
108
109 Nat
Fig. 3. The enthalpy of formation for graphite as the function of the number of atoms in a nanocrystal for its various shapes: (1) fiber with eccentricity Е = 10, (2) sheet with eccentricity Е = 0.1, (3) compact shape of graphite nanocrystal, and (4) thermodynamically favorable graphite nanocrystal shape.
surface energy attains the value of ≤10—2 at the eccentricity that allows considering this prism to be the film. The second case is the equilibrium between nanodiamond and nanographite of a compact shape for which the most distant points—vertexes—are at a minimum distance from the center to the definite number of atoms in crystal. Such a shape is the prism with eccentricity E = 2/2 . In the third situation we analyze the equilibrium of carbon phases when the solid phases are nanodisper-
sive elongated crystals; i.e., prisms with large equal eccentricity values (fibers). This is a case when the nanographite–nanodiamond equilibrium pressure may decrease when compared to the pressure at the phase diagram of the typical bulk-crystalline carbon. We estimated therefore the change in standard enthalpy of formation of nanographite in the form of a hexagonal prism and diamond in the forms of an octahedron and a hexagonal prism. As is seen from Fig. 2, the abundant values of ° ) tend to enthalpy of nanoparticle formation Δ( Δ f H 198 zero with an increase in characteristic size of the diamond nanoparticles irrespective of their shapes. At the same time, the nanocrystals shape defines the abundant enthalpy value attained at the definite characteristic size of nanoparticle. The enthalpies of diamond formation as functions of characteristic nanocrystals size with the same thermodynamically favorable shape of nanoparticles but different internal structures of nanodiamond are shown in Fig. 2 by lines 4 and 5 for the cubic and hexagonal structures of nanocrystals, respectively. As is seen, not only the shape and size of the diamond nanoparticle, but also the internal structure of carbon-atom arrangement in nanocrystal ° ). affects the value Δ( Δ f H 198
As is observed from Fig. 3, the abundant enthalpy of the graphite and diamond nanoparticle formation depends on both the number of carbon atoms and their ° ) exhibits a shape. Simultaneously, the value Δ(Δ f H 198 more pronounced dependence on the nanoparticle size and increases abruptly with the decrease in the characteristic size of carbon nanocrystal. On basis of the EoS for graphite and diamond with corrections in enthalpy of formation and entropy as functions of size and shape of nanoparticles, one can plot the phase diagrams of nanocarbon particles for different sizes, shapes, and internal structures of atom arrangement in the nanoparticle from the conditions of thermodynamic phase equilibrium (equality of chemical potentials) (Fig. 4). The calculations reveal that the nanographite– nanodiamond equilibrium pressure increases with the decrease in the number of atoms in crystal (with increasing dispersivity) (Fig. 4). For a thermodynamically favorable shape of crystals, as well as for the compact shape of cubic nanodiamond, the nanographite–nanodiamond equilibrium lines are displaced upwards with increasing dispersivity due to the growth of difference between the changes in the abundant enthalpy of formation of nanodiamond and nanographite ΔНdiam_graph. The references [1, 17–20], which are dedicated to experimental and calculated dates of nanodiamond formation in detonation products, also denote the increase in the pressure of nanographite–nanodia-
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p, GPa 25 1 8 6 2 7 3
20 15
4 5
10 5
0 300 800 1300 1800 2300 2800 3300 3800 4300 4800 T, K Fig. 4. The pressure of diamond–graphite phase equilibrium as the function of temperature for nanocrystals in the form of regular hexagonal prism for diamond and graphite: (1) E = 1, Nat = 104; (2) E = 1, Nat = 106; (3) E = 10, Nat = 105; (4) E = 100, Nat = 105, (5) E = 200, Nat = 105; (6) for crystals of cubic diamond and hexagonal graphite Nat = 105, (7) equilibrium line [3] of bulk-crystalline carbon; and (8) equilibrium line for the thermodynamically favorable shape of nanoparticles of hexagonal diamond and graphite Nat = 105.
mond phase equilibrium with decreasing sizes of carbon nanocrystals. As is obvious from Fig. 4, the diamond–graphite phase diagram for nanocrystals of hexagonal graphite and cubic diamond in the form of regular hexagonal prism with the same number of fiberlike arranged atoms (Nat = 105, lines 3, 4 and 5) are shifted to the lower pressures with increasing eccentricity (lines 3, 4, and 5). For diamond and graphite nanoparticles with the same shape and the definite eccentricity (E = 1) in the form of a hexagonal prism, the height of which is equal to the diameter of the circumscribing circle around the hexagon in the base, the phase equilibrium lines are shifted to the phase equilibrium of typical bulk-crystalline diamond and graphite (line 7) with an increasing number of atoms (here, line 1 corresponds to Nat = 10 4, and line 2—Nat = 106). The position of graphite–diamond phase equilibrium line is defined not only by the size and shape of nanoparticles, but also by the atom arrangement in nanocrystal. For particles in thermodynamically favorable shapes with the same number of atoms Nat = 105 but varying internal structures of atom arrangement (line 6 corresponds to the cubic diamond and line 8 corresponds to the hexagonal one), the position of equilibrium lines is different. NANOTECHNOLOGIES IN RUSSIA
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The key role in the position of phase equilibrium lines is played by the difference between the changes in enthalpy of formation of different phases [10]:
° ) − Δ(Δ H ° ) . (14) Δ H 12 = Δ(Δ f H 298 1 f 298 2 Three cases are thus implemented independently on the absolute values Δ(Δ H ° ) of each phase: f
298
ΔH12 > 0: the equilibrium pressure is shifted to Phase 1 compared to its value for the same nondispersive phases; ΔH12 = 0: the equilibrium pressure is not shifted; ΔH12 < 0: the equilibrium pressure is shifted to Phase 2. ° ) is calculated The value of the correction Δ(Δ f H 298 for crystal depending on its size. The position of phase equilibrium line is affected by the difference between the changes in entropy of different carbon phases: (15) Δ S12 = Δ S1 − Δ S 2. Regardless of the absolute values ΔS1 and ΔS2, three situations can be considered: ΔS12 > 0: the shift in the equilibrium pressure to Phase 1 decreases with increasing temperature (the change in the tangent angle of the equilibrium line increasing in case of diamond–graphite equilibrium) compared to the equilibrium pressure of the same carbon phases in a typical bulk-crystallite state; ΔS12 = 0: no shift for the equilibrium pressure is observed; ΔS12 < 0: the shift of the equilibrium pressure to Phase 1 increases with increasing temperature (the change in the tangent angle of the equilibrium line decreasing in case of diamond–graphite equilibrium). As is proven by the calculations, unlike enthalpy, the changed entropy exerts less influence on the equilibrium pressure of diamond–graphite nanoparticles and on the triple-point position. Consequently, in order to estimate the position of phase equilibrium line, the change in enthalpy of the nanocarbon particle formation has first of all to be taken into account. As is seen from Fig. 5, the absolute value of difference between enthalpies of formation for cubic nanodiamond and hexagonal nanographite increases with a decreasing number of atoms in nanocrystal. The increase in eccentricity Е ≥ 2 leads to negative values of difference between enthalpies of formation for cubic nanodiamond and hexagonal nanographite ΔHdiam–gr ≤ 0, which corresponds to the decrease in the phase equilibrium pressure of diamond–graphite nanoparticles respective to the equilibrium of bulk-crystalline structures of diamond and graphite. Note that in case of very elongated fiberlike crystals (eccentricity E = D 1), the equilibrium lines are 2B 2015
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ΔH, kcal/mol 4 3
1
2 1 0 2 10 −1 −2 −3 −4
2 103
104
105
106
3 4 5 6
−5 Fig. 5. The difference between the changes in enthalpy of formation for cubic nanodiamond and hexagonal nanographite as the function of the number of atoms in crystal
° )diam – Δ(Δ f H 298 ° ) gr at Е > 1: ΔHdiam–gr = Δ(Δ f H 298 (1) E = 1, (2) E = 2, (3) E = 4, (4) E = 6, (5) E = 8, and (6) E = 10. shifted to the low pressures, as is seen in Fig. 4 (the decrease in the graphite–diamond transition temperature for a fixed pressure with decreasing nanoparticle size). The executed calculations reveal thus that each thermodynamic phase containing nanodispersive particles with the same size, shape, and structure has its own phase diagram with the proper positions of equilibrium lines and triple point. REFERENCES 1. A. L. Vereshchagin, Detonation Nanodiamonds, Morphology (Polzunov Altai State Techn. Univ., Biisk, 2001), p. 177 [in Russian]. 2. N. D. Orekhov and V. V. Stegailov, “Moleculardynamical simulation of graphite melting,” Teplofiz. Vys. Temp. 52 (2), 220 (2014). 3. F. P. Bundy, “The P,T phase and reaction diagram for elemental carbon,” Geophys. Res. 85 (12), 6930 (1980). 4. M. Togaya, S. Sugiyama, and E. Mizurhara, Melting Line of Graphite. High Pressure Science and Technology Ed. by S. C. Schmetd, J. W. Chaner, and M. Ross (Am. Inst. Phys., 1993), pp. 255–258. 5. A. Yu. Basharin, I. G. Lippgardt, and M. Yu. Marin, “Boiling and melting of quasi-monocrystalline graphite,” Trudy Inst. Teplofiz. Ekstrem. Sost. Ob”ed. Inst. Vys. Temp. 3, 79 (2000). 6. V. N. Korobenko, A. I. Savvatimski, and R. Cheret, “Graphite melting and properties of liquid carbon,” Int. J. Thermophys. 20 (4), 1247–1256 (1999). 7. L. F. Vereshchagin and N. S. Fateeva, “Melting temperatures of refractory metals at high pressures,” High Temp. – High Press. 9 (6), 619–628 (1977).
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