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J. Phys. Chem. B 2002, 106, 6634-6637
ARTICLES Kinetics of Diamond Crystallization from the Melt of the Fe-Ni-C System Vladimir L. Solozhenko,* Vladimir Z. Turkevich, and Oleksandr O. Kurakevych Institute for Superhard Materials of the National Academy of Sciences of Ukraine, 04074 KieV, Ukraine
Wilson A. Crichton and Mohamed Mezouar European Synchrotron Radiation Facility, 38043 Grenoble, France ReceiVed: July 26, 2001; In Final Form: March 27, 2002
X-ray powder diffraction with synchrotron radiation was used for the first time to study in situ diamond crystallization from the Fe-Ni-C melt at pressures up to 6 GPa and temperatures up to 1700 K. At 5.2 GPa over the whole temperature range of diamond crystallization (1510-1605 K), the melt is in equilibrium with both diamond and fcc Fe-Ni-C solid solution (γ-phase); that is, the L ) C + γ monovariant eutectic reaction takes place. From the non-isothermal kinetic data, it follows that diamond crystallization is controlled by carbon diffusion in the melt. Kinetic data are best fitted by the model that assumes a constant nucleation rate and a three-dimensional growth of nuclei, with an apparent activation energy of 148(64) kJ mol-1.
Introduction The strength of diamond crystals depends on the crystal structure perfection, which in turn is defined by the growth kinetics. Therefore, to produce diamonds with predetermined properties, for example high strength and thermal stability, knowledge of kinetic regularities of crystal nucleation and growth in transition metal-graphite systems that are traditionally used for the commercial synthesis of diamond is required. A majority of publications report the kinetics of diamond crystal seed growth,1-4 while publications on diamond spontaneous crystallization report only the effect of thermodynamic parameters on a number of nuclei and the degree of graphiteto-diamond conversion5,6 as well as the effect of the structure of the initial carbon material on the conversion degree and size of the grown diamond crystals.7-9 In all aforementioned papers, quenching from high temperatures and pressures has been used. This resulted in insufficient accuracies of evaluation of the initial pressure and temperature, variations of p,T-parameters in the course of crystallization, the uncertainty of determination of the instant of crystallization, and the comparative periods of time required to reach the operating condition relative to the rate of the crystallization process. Hence, it is clear that reliable data on the kinetics of diamond crystallization can be obtained only in situ using X-ray diffraction. However, studies of low-Z elements, specifically of carbon, at high pressures and temperatures in real time are only possible when high-intensity synchrotron-derived X-ray radiation from third-generation sources is used. In the present work, the kinetics of diamond spontaneous * Corresponding author. Postal address: Institute for Superhard Materials of the National Academy of Sciences of Ukraine, 2, Avtozavodskaya St., Kiev 04074, Ukraine. Telephone: +380 44 430 25 78. Fax: +380 44 468 86 25. E-mail:
[email protected].
crystallization from the melt of the Fe-Ni-C system has been studied in situ for the first time using X-ray diffraction with synchrotron radiation at the European Synchrotron Radiation Facility. Experimental Section The Fe60Ni40C20 alloy was melted in a corundum crucible in an induction furnace in an argon atmosphere using iron (99.5%), nickel (99.8%), and high-purity synthetic graphite (99.999%, -325 mesh) and then cast into a copper mold. The carbon content of the alloys was determined by quantitative chemical analysis. Prior to the experiments on diamond crystallization, the alloy was crushed using cemented carbide mortar and mixed with graphite powder in the 1:4 molar ratio. The average size of the alloy particles was 120 µm, while that of graphite was 250-300 µm. The high-pressure experiments were carried out using a Paris-Edinburgh press at beamline ID30, ESRF. The mixed sample powder was subsequently loaded into a capsule of hexagonal graphite-like boron nitride (hBN) and included in a cylindrical graphite furnace. A boron-epoxy gasket was used as the compressing medium between the conical tungsten carbide anvils. Energy-dispersive data were collected on a Canberra solidstate Ge-detector with a fixed Bragg angle 2θ ) 10.03° using a white beam from the wiggler collimated to 70 µm × 100 µm (horizontal by vertical) and the detector optics set with a 2θ acceptance angle of 0.005°, which ensures high resolution of the observed diffraction patterns. The collection time was 20 s for each diffraction pattern. The detector was calibrated using the KR and Kβ fluorescence lines of Cu, Rb, Mo, Ag, Ba, and Tb. The sample temperature was controlled with a calibrated computer-controlled dc power supply driven with a 0.002 V
10.1021/jp012899s CCC: $22.00 © 2002 American Chemical Society Published on Web 06/06/2002
Kinetics of Diamond Crystallization
J. Phys. Chem. B, Vol. 106, No. 26, 2002 6635
Figure 1. Region of diamond spontaneous crystallization in the Fe-Ni-C system (diamond forms within the BOD triangle, viz. in the p,T-region restricted by the dashed curve). The CD-line is the melting curve of the γ + Fe3C + C ternary eutectic according to our in situ experiments (solid squares are the experimental points). AB is the graphite-diamond equilibrium line.13 Constant pressure heating paths are delineated by dashed arrows.
step resolution over its 0-5 V range (proportional to the 0-200 A supply). Pressures and temperatures were estimated by analytically cross-calibrating the equations of state of hBN and Pt with the program PTX-cal10 using the unit-cell volumes obtained by least-squares fits of the ED diffraction patterns obtained in situ. The relatively high contrast between the compressibilities and thermal expansivities of hBN11 and Pt12 ensures that errors of 0.2 GPa and 20 K in pressure and temperature can be realized.
Figure 2. Diffraction patterns of the Fe-Ni-C system taken at 5.2 GPa at the heating rate 25 K/min.
Results and Discussion Figure 1 shows the p,T-region of diamond crystallization in the Fe-Ni-C system relative to the graphite-diamond equilibrium line13 and the melting line of the γ + Fe3C + C ternary eutectic14 defined in the present work. It should be noted that the literature data on the melting of the ternary eutectic in the Fe-Ni-C system at high pressures and temperatures are rather contradictory. According to Kocherzhinsky et al.,15 the melting temperature of the ternary eutectic (Te) at 6 GPa is 1400 K, while, according to Pavel et al.,16 Te ) 1665 K at 5.7 GPa. We have found that in the 4-6 GPa range the pressure dependence of the melting temperature of the ternary eutectic is described by the following equation
Te [K] ) 1346(27) + 30(6)p [GPa]
(1)
The experiments in ref 16 were conducted by the quenching method; therefore, the errors in the temperature determination are potentially serious. As to the paper in ref 15, the authors have not evidently managed to avoid the interaction between the samples and a thermocouple when they conducted the DTA experiments under pressure. Because of this, they have attributed the thermal effect observed at 1400 K to the melting of eutectics, which has resulted in an understated (by more than 100 K) evaluation of the position of the melting line of the γ + Fe3C +C ternary eutectic. At 4.1, 4.6, and 5.0 GPa, diamond lines are not observed in diffraction patterns up to 1700 K; that is, below 5.0 GPa the diamond crystallization from the Fe-Ni-C melt does not occur because a liquid phase appears in the system only in the p,Tregion of the thermodynamic stability of graphite (4.1 and 4.6 GPa) or just 10-20 K below the graphite-diamond equilibrium line (5.0 GPa).
Figure 3. Polythermal section C-60%Fe,40%Ni of the Fe-Ni-C ternary system at 5.2 GPa.15
At 5.2 GPa, using a linear heating rate of 25 K/min, diamond crystallization starts at 1510 K, immediately after the broad line corresponding to the liquid phase appears in the diffraction patterns, and is fully completed at 1605 K (Figure 2). We note that despite the presence of a liquid in the system over the whole temperature range of diamond crystallization, in parallel with lines of diamond and graphite, the diffraction patterns exhibit a line at E ≈ 39.4 keV (d ≈ 1.8 Å) that can be ascribed to the fcc Fe-Ni-C solid solution. This fact indicates that in accordance with the p,T-phase diagram of the Fe-Ni-C system15 (Figure 3) diamond crystallization occurs in the L + D + γ three-phase region where the melt is in equilibrium with both diamond and the γ-phase. The existence of the L + D + γ three-phase region in the C-60%Fe,40%Ni polythermal section, which is given in Figure 3, is due to the fact that the section plane intersects the line of the L ) D + γ monovariant eutectic reaction. According to the Gibbs phase rule, between two (L + D and D + γ) twophase regions in the polythermal section the L + D + γ threephase region is located. The temperature interval in which this
6636 J. Phys. Chem. B, Vol. 106, No. 26, 2002
Solozhenko et al. Knowing the linear heating rate, b ) dT/dt, we pass to the differential of the temperature on the right side
d{[-ln(1 - R)]1/r} )
k(T) dT b
(5)
Relation 5 is the generalization of eq 2 for non-isothermal conditions. The rate constant can be represented as
k*(T) ) K′(T)[D(T)]1.5(∆c)1.5
Figure 4. Degree of graphite-to-diamond conversion versus temperature plot for the diamond crystallization from the Fe-Ni-C melt at 5.2 GPa and the heating rate 25 K/min. The solid line is the calculated kinetic curve (r ) 2.5).
region exists has been established in the present work. As all the diffraction patterns exhibited diffraction lines of graphite (Figure 2), we state that diamond crystallization from the Fe-Ni-C melt occurs at constant carbon supersaturation with respect to diamond, which is ensured by the dissolution of the initial graphite. On the basis of this, the degrees of the graphiteto-diamond conversion proceeding via melt have been calculated by normalizing the integral intensities of the (111) reflection of diamond at various temperatures to the appropriate value at 1605 K (R ) 1). The results obtained at 5.2 GPa and the heating rate of 25 K/min are shown in Figure 4 as the degree of the graphite-todiamond conversion (R) versus temperature. Diamond crystallization in the system under study is very fast and proceeds in a narrow temperature range that essentially plagues isothermal kinetics studies. Because of this, the present work describes a non-isothermal approach. However, even in a non-isothermal experiment, we managed to obtain only a limited number of points on the kinetic curve (Figure 4). In this case, traditional methods of calculating kinetic parameters are unsuitable. The inverse problem on non-isothermal kinetics requires a method of searching for a solution using classes of correctness (that is, on sets, where the problem changes to the well-posed one) with a subsequent test of the obtained solution for consistency with the criteria given below.17 One of the advantages of this method is that it is possible to construct the solution on the basis of a small number of experimental points. As the set of the possible solutions, we have taken a relation that fits the generalization of the Avrami equation for nonisothermal conditions. The initial Avrami equation for isothermal kinetics is of the form18
-ln(1 - R) ) k*tr
(2)
where R is the degree of transformation, k* is the constant of the crystallization rate, and r is the order of the kinetic equation. Assuming that the form of the equation holds in the temperature range under study, we have
[-ln(1 - R)]1/r ) k(T)t
(3)
We take differentials of the right and the left sides of the equation
d{[-ln(1 - R)]1/r} ) k(T) dt
(4)
(6)
where K′(T) is the nucleation rate constant, D is the coefficient of volume diffusion, and ∆c is the difference between graphite and diamond solubilities in the melt. In our calculations, we have used the temperature dependence of ∆c at 5.2 GPa found by processing the data reported in ref 19
∆c ) 2.89 × 10-2 - 1.77 × 10-5T
(7)
Equation 6 describes the dependence of the diamond growth rate on supersaturation (∆c), which in turn depends on temperature and decreases as the graphite-diamond equilibrium line is approached (see eq 7). Thus, introduced dependencies allow us to take into account the variation of the difference in solubilities between graphite and diamond with the temperature in the models for non-isothermal kinetics and their effect on the diamond crystal growth. Equation 5 has been used in the form
[ (∫
R ) 1 - exp -
Tk(T)
T0
b
)] r
dT
(8)
assuming that the temperature dependence of the rate constant is of the Arrhenius type
k(T) ) e(z-Ea/RT)(∆c)1.5/r
(9)
where Ea is the apparent activation energy of crystallization (J mol-1 K-1) and z is the logarithm of the preexponential factor for the rate constant in the Arrhenius equation. The following requirements have been imposed on the desired solution: (1) a small change in the initial data (within the maximal error) should correspond to a small change in the solution (continuity); (2) the discrepancy functional that fits the desired solution should not exceed the lowest possible value by more than 20%; (3) the solution has a real physical meaning (this refers especially to r). Of all the possible r ∈ [1;4], only r ) 2.5 meets all the above requirements. For the other r-values being considered (1.0, 1.5, 2.0, 3.0, 3.5, and 4.0), both an increase in the discrepancy functional (16-65% of the minimal value for r ) 2.5) and the loss of solution stability are observed. For r ) 2.5, we have obtained the following kinetic parameters:
Ea ) 148 ( 64 kJ mol-1 (z - ln b) ) 11.4 ( 5.0 which are the unique stable solution of the initial malposed kinetic problem. According to ref 20, the r ) 2.5 value fits the mechanism, which assumes a constant nucleation rate and three-dimensional growth of the resulting nuclei under the conditions where the limiting stage of the crystallization is the carbon diffusion to the surface of a growing diamond crystal.
Kinetics of Diamond Crystallization The apparent activation energy of the diamond crystallization from the melt of the Fe-Ni-C system found in the present work is in surprisingly good agreement with the corresponding value 140 kJ‚mol-1 evaluated from quenching data for the NiMn-C system at 6 GPa and 1520-1610 K.21 This fact supports the view that both Fe-Ni-C and Ni-Mn-C melts belong to the so-called conventional solvents, that is, those whose melting immediately results in the diamond crystallization as opposed to nontraditional solvents (for example Mg-Zn-C22) where a considerable (up to 800 K) increase in temperature upon melting is needed to ensure the diamond nucleation. Thus, for the Fe-Ni-C system we have found that the process, which restricts the growth of a diamond nucleus over the whole range of supersaturation, is diffusion and that the three-dimensional growth of the nuclei takes place. Hence, the rate of diamond crystal growth might be controlled by regulating the intensity of diffusion mass transfer through variations either in the degree of supersaturation (offset from the graphitediamond equilibrium line) or in the composition of the crystallization medium (change in the carbon diffusion coefficient). According to ref 23, at 5.35 GPa and 1430 K a number of centers of diamond crystallization from the melt of the NiMn-C system increase avalanche-like for 1-3 min. At this point, the nucleation rate decreases drastically, and after a lapse of 480-600 s, the nucleation of crystals is unobservable. Under isothermal conditions, the time needed for the nucleation sites to be exhausted essentially depends on the pressure and varies from 600 s at 4 GPa to 5 s at 8 GPa.24 This is attributable to an increase of the difference in chemical potentials between graphite and diamond, which is the thermodynamic impetus of the transformation, as well as to an increase of the nucleation rate with pressure. As demonstrated by Figure 2, in the present work the graphite-to-diamond conversion has been fully accomplished between 1510 and 1605 K at the heating rate 25 K/min, that is, within 4 min. Evidently, the conversion degree R ) 1 has been achieved before the nucleation sites have been exhausted. This is responsible for the fact that the best description of our experimental data can be made under the assumption that the nucleation rate is constant. Conclusions In this paper we have illustrated how the use of in situ X-ray diffraction with synchrotron radiation provides an increasingly detailed picture of diamond crystallization and an excellent method of following fast crystallization kinetics in real time.
J. Phys. Chem. B, Vol. 106, No. 26, 2002 6637 This is particularly important for the rapid crystallization we have observed here where quenching studies would be most difficult to perform. Our study has highlighted important features of diamond crystallization in conventional growth systems: the process is diffusion-controlled and is characterized by a constant nucleation rate and three-dimensional growth of nuclei. Acknowledgment. The authors thank Mr. N. Guignot for assistance in high-pressure experiments and Dr. D. Andrault for helpful discussions. This work was carried out during beam time allocated to Proposal CH-985 at ID30, ESRF. References and Notes (1) Strong, H. M.; Hanneman, R. E. J. Chem. Phys. 1967, 46, 3668. (2) Strong, H. M.; Wentorf, R. H. Naturwissenschaften 1972, 59, 1. (3) Vagarali, S.; Lee, M.; Devries, R. C. J. Hard Mater. 1990, 1, 233. (4) Novikov, N. V.; Ivakhnenko, S. A.; Katsay, M. Ya. New Diamond Science and Technology, Proceedings of the 2nd International Conference; MRS: Pittsburgh, 1990; p 71. (5) Litvin, Yu. A. Growth of Crystals; Nauka: Moscow, 1972; Vol. 9, p 65 (in Russian). (6) Bezrukov, G. N.; Butuzov, V. P.; Laptev, V. A. Dokl. Akad. Nauk SSSR 1971, 200, 1088 (in Russian). (7) Wentorf, R. H. J. Phys. Chem. 1965, 69, 3063. (8) Kasatochkin, V. I.; Shterenberg, L. E.; Slesarev, V. N.; Nedoshivin, Yu. N. Dokl. Akad. Nauk SSSR 1970, 194, 801 (in Russian). (9) Tselikov, A. I.; Krylov, V. S.; Gankevich, L. T.; Zyuzin, V. I. Dokl. Akad. Nauk SSSR 1982, 265, 681 (in Russian). (10) Crichton, W. A.; Mezouar, M. High Temp.sHigh Pressures 2002, 34, 235. (11) Solozhenko, V. L.; Peun, T. J. Phys. Chem. Solids 1997, 58, 1321. (12) Holmes, N. C.; Moriarty, G. R.; Gathers, G. R.; Nellis, W. J. J. Appl. Phys. 1989, 66, 2962 and references therein. (13) Bundy, F. P. Physica A 1989, 156, 169. (14) γ-fcc Fe-Ni-C solid solution (γ-phase). (15) Kocherzhinsky, Yu. A.; Kulik, O. G.; Turkevich, V. Z. High Temp.sHigh Pressures 1993, 25, 113. (16) Pavel, E.; Pintiliescu, L.; Baluta, G.; Giurgiu, C.; Barb, D.; Lazar, D. P.; Morariu, M. Physica B 1991, 175, 354. (17) Arsenyuk, V. Ya. Methods of Mathematical Physics and Special Functions; Nauka: Moscow, 1984 (in Russian). (18) Avrami, M. J. Chem. Phys. 1939, 7, 1103; 1940, 8, 212; 1941, 9, 177. (19) Turkevich, V. Z.; Kulik, O. G. High Pressure Res. 1995, 14, 175. (20) Sˇesta´k, J. Thermophysical Properties of Solids. Their Measurements and Theoretical Thermal Analysis; Academia: Prague, 1984. (21) Katsay, M. Ya. Cand. Sci. (Eng.) Thesis, ISM, Kiev, 1986 (in Russian). (22) Shulzhenko, A. A.; Ignat’eva, I. Yu.; Osipov, A. S.; Smirnova, T. I.; Podzyarey, G. A.; Belyavina, N. N.; Markiv, V. Ya. Synthesis, Sintering and Properties of Superhard Materials; ISM: Kiev, 2000; p 15 (in Russian). (23) Khadzhi, V. E.; Tsinober, L. I.; Shterenlikht, L. M.; et al. Synthesis of Minerals; Nedra: Moscow, 1987 (in Russian). (24) Shulzhenko, A. A. Dr. Sci. (Eng.) Dissertation, ISM, Kiev, 1990 (in Russian).
