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Stability of hexagonal modification of fullerite C60
Carbon 43 (2005) 803–808 www.elsevier.com/locate/carbon
Stability of hexagonal modification of fullerite C60 E.V. Skokan *, I.V. Arkhangelskii, D.E. Izotov, N.V. Chelovskaya, M.M. Nikulin, Yu.A. Velikodnyi Physical Chemistry Department, Moscow State University, VorobÕevy Gory, 119899 Moscow, Russia Received 17 September 2004; accepted 3 November 2004 Available online 8 December 2004
Abstract Stability of the hcp modification of fullerite C60 under heating, compression, and mechanical grinding was studied. It was established that uniaxial load and mechanical grinding caused transition of the hcp modification into more stable fcc modification, while no transition was observed upon heating or hydrostatic pressure. The results were discussed in terms of the theory of the deformation of solids. 2004 Elsevier Ltd. All rights reserved. Keywords: A. Fullerene; B. Arc discharge; C. X-ray diffraction; D. Crystal structure
1. Introduction Fullerite C60 comes in three polymorphic modifications, namely, simple cubic (sc), face-centered cubic (fcc), and hexagonal close-packed (hcp) modifications [1–3]. Cubic and hexagonal close-packed structures can be thought off as being polytypic modifications, since only the packing fashions of identical layers are different in their crystal lattices (sphalerite and wurtzite lattices). The cubic lattices consist of three layers, while the hexagonal ones have two layers. Unlike fcc structures, hcp (wurtzite) structures made of similar atoms are usually unstable under normal conditions and seldom seen in practice [4]. In the case of fullerite C60, the hcp modification is metastable with respect to the stable fcc modification. Therefore, it can be synthesized under undoubtedly non-equilibrium conditions, for example, upon crystallizing from fullerene solutions frozen with liquid nitrogen [5]. Note that cubic modifications of C60 have been studied extensively, while the properties
*
Corresponding author. E-mail address: [email protected] (E.V. Skokan).
0008-6223/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2004.11.007
of the metastable hexagonal modification as an individual compound are poorly understood. The aim of this work was to study the stability of the hcp modification of fullerite C60 under heating, compression, and mechanical grinding.
2. Experimental 2.1. Materials The samples of hcp modification were obtained by extraction from the saturated benzene solution of C60 (fcc) frozen with liquid nitrogen; hexane was used as an extractant [5]. The initial sample of C60 (fcc) contained C60 > 99.98%, C60O and C70 < 0.02%, as established by HPLC. Prior to use, all organic solvents (HPLC purity grade) were distilled. The extracted product was air dried and heated under vacuum at 700 K for 3 h. According to X-ray diffraction date (Fig. 1), its structure was identified as hcp with the following lattice parame˚ , c = 16.38 A ˚ , which are in good agreeters: a = 10.02 A ment with our previous results [3,5].
113
(a)
112
110
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220
E.V. Skokan et al. / Carbon 43 (2005) 803–808
111
804
10
20
201
114 015 205 222,123 006
121
102
004
101
103
002
100
(b)
30
40
50
60
Fig. 1. X-ray powder diffractograms of (a) fcc and (b) hcp phases of fullerite C60.
The content of the hcp phase in the sample obtained was estimated to be 95% on the basis of the ratio of intensities of the h1 0 3i and h1 1 0i lines taken from the X-ray diffraction pattern. The sample, in which the content of the hcp phase was previously established by the Rietweld method [3], was used as a standard. The amount of residual solvents (0.01–0.02%) was determined from H1 NMR spectra (a Bruker AC-200 spectrometer, room temperature) as in [6]. To identify the solvents, IR spectra of the samples obtained were measured (a Specord M-80 instrument) within the range of 400–4000 cm1; the samples were used in the form of pellets with KBr. No vibrations of the C–H bonds were observed in the IR spectra. 2.2. Apparatus and procedures The material obtained was heated at different temperatures, compressed (uniaxial load and hydrostatic compression) and mechanically ground in a mortar; the content of fcc and hcp modifications after the treatments was monitored at normal conditions (room temperature, p = 1 atm) using X-ray diffraction analysis (a STUDI/P diffractometer, CuKa1 radiation, Ge monochromator, a PSD detector). The reflection with hkl h1 0 3i, which may be observed only for the hcp modification but is forbidden for the fcc, was used for estimating stability of the hcp modifi-
cation, since the change in the intensity of the h1 0 3i line is proportional to the change in the content of the hcp modification in the sample. The description of the line profile was improved by the profile analysis of the X-ray diffraction patterns shown in Fig. 1 that made it possible to consider the reflections absent in the diffraction pattern. All calculations were performed using the WinXPOW program package applied to a STUDI diffractometer. The relative volumes of the samples at different pressures were measured as in [7]. In each experiment, four different weight portions of fcc and hcp samples were used; the pressure changed from 0 to 2.6 GPa with the step of 0.05 GPa. Compression of the samples was studied in a Carver laboratory press using stainless piston-cylinder molds. Mechanical grinding was performed in an agate mortar for 30 min. To study stability of the hcp modification under compression, the sample was pelleted and pellets (30.0 ± 0.5 mg) were subjected to uniaxial loading. A series of experiments was carried out at different pressures (0.3, 0.5, 0.8, 1, 1.6, and 2.7 GPa) but similar time periods (60 min). Another series was performed at similar pressure (0.8 GPa) but different time periods (5, 15, 30, 45, and 60 min). After each run, the load was relieved, and X-ray diffraction analysis of the pellets was carried out. To study the effect of hydrostatic pressure on the hcp sample, the weight portion (ca. 30 mg) was charged into an aluminum ampoule filled with ethanol. The ampoule was shaked for eliminating gas bubbles, then covered with a lid and placed into a press mould. Once the experiment was finished, the product was dried and analyzed by X-ray diffraction method.
3. Results and discussion 3.1. Heating According to X-ray analysis, no phase transition took place upon heating the samples at 750 K for 5 h (Fig. 2). At elevated temperature, the X-ray diffraction patterns showed the decrease in peak intensities with increasing background. At 1200 K, no reflections were observed that is evidence of destruction of the carbon cage of the fullerite molecule. 3.2. Uniaxial load The relative volumes (V/V0) of the fcc and hcp samples vs. load are shown in Fig. 3. The curves obtained for both samples are similar. The X-ray diffraction analysis showed that the fcc samples stayed unchanged after experiments, while the hcp structure transformed into
E.V. Skokan et al. / Carbon 43 (2005) 803–808
805
35
1200 K
30
25
1100 K 20
15
900 K
10
5 0.0
0.5
1.0
1.5
2.0
2.5
3.0
Fig. 4. Intensity of the h1 0 3i peak for hcp sample vs. load.
750 K
5
10
15
20
25
30
35
Fig. 2. X-ray powder diffractograms of the hcp phase after thermal treatment at different temperatures.
fcc structure. The compression curves are smooth and monotonous, thus indicating that the hcp–fcc transition is not discontinuous but goes on within a wide pressure interval; therefore, it can be classified as an martensitetype transition. It should be noted that transition between lonsdelite and diamond, belonging to polymorphic carbon modifications, also proceeds continuously [4].
The extent of transformation of hcp modification into fcc depending on pressure is illustrated in Fig. 4. As seen from this figure, the hcp–fcc transition proceeded within a wide pressure range that is in good agreement with the above data. Note that everywhere over the range studied (up to 2.7 GPa), the hcp modification was not completely transformed into the fcc modification in the timescale of our experiments (Fig. 5). The time dependence of the hcp–fcc transformation at constant load of 0.8 GPa (Fig. 6) can be closely approximated by the equation a 0.07 · t0.45±0.05, where a is the extent of transformation and t is the time. It is rather similar to the time dependence of the martensite-type phase transitions [8]. 3.3. Hydrostatic pressure The sample was compressed at 0.5 GPa for 60 min; however, X-ray diffraction analysis showed no change
1.00
1.00
(b)
(a) 0.98
0.98
0.96
0.96
0.94
0.94
0.92
0.92
0.90
0.90
0.88
0.88 0.0
0.5
1.0
1.5
2.0
2.5
0.0
0.5
1.0
1.5
Fig. 3. The relative volume of (a) fcc and (b) hcp phases vs. load.
2.0
2.5
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E.V. Skokan et al. / Carbon 43 (2005) 803–808
(b)
(c)
(b)
(a)
(a)
5
10
15
20
25
10
Fig. 5. X-ray diffractograms of the hcp samples: (a) initial, (b) subjected to uniaxial load of 0.8 GPa, and (b) 2.7 GPa.
20
30
40
50
60
70
Fig. 7. X-ray diffractograms of the hcp samples: (a) initial, (b) subjected to hydrostatic pressure of 0.8 GPa for 100 min.
3.4. Mechanical grinding
0.5
According to X-ray diffraction data (Fig. 8), grinding results in reducing content of the hcp modification in the sample. Thus, the hcp–fcc transition takes place under mechanical grinding.
0.4
0.3
3.5. Hcp–fcc transformation in fullerite C60 in terms of the deformation theory
0.2
0.1
0.0 0
10
20
30
40
50
60
Fig. 6. The time dependence of the hcp–fcc transformation at constant uniaxial load of 0.8 GPa. The solid line is a least-square fit: a 0.07 · t0.45.
in the sample. The increase in pressure and time period (to 0.8 GPa and 100 min, respectively) virtually did not affect the X-ray diffraction pattern (Fig. 7): only slight change in the ratio of peak intensities was observed. Thus, no hcp–fcc transition took place under hydrostatic pressure.
To understand behaviour of the hcp phase of C60 under mechanical treatment, some aspects of the deformation theory of solids should be engaged. The hcp–fcc transition is accompanied by parallel shift of hexagonal layers (Fig. 9). Let direct the Oz axis across the layers, and the Ox, Oy axes direct symmetrically into the layer plane. When layer A shifts with respect to layer B in the direction shown in Fig. 9, then the strain tensor takes the form 0 1 0 0 j 1B tan a C e ¼ @ 0 0 j A, j ¼ pffiffiffi , 2 2 j j 2j2 where a is the slope angle of the Oz vector. In the case of the hcp–fcc transition, j = 1/2, that corresponds to ductile deformation. However, to evaluate the stress required for initiating this deformation, the linear theory
E.V. Skokan et al. / Carbon 43 (2005) 803–808
Thus, to realize the deformation under study, tangent stresses are required (p13 and p23 do not vanish). Obviously at hydrostatic pressure the stress tensor is spherical over all space, i.e. tangent stresses are absent, and the hcp–fcc transition is impossible. When compression is concentrated along one axis, then
(b)
pij ¼ pli lj ,
10
15
20
25
30
ð2Þ
where l is the unit vector along which the load p is applied. Hence it is evident that anisotropic compression causes non-vanishing tangential stresses. A close look at equations (1) and (2) shows that the stress tensors do not coincide for any l; therefore, in the case of single crystal, the hcp–fcc transition cannot be observed at uniaxial compressive load. In polycrystalline samples, such a coincidence is possible, in principle, since in the latter case, distribution of stresses on the surface and, naturally, in a bulk, is different. However, it is not the fact in the case of hydrostatic pressure. Really, the particles of a sample (even powder) are dispersed in a liquid; therefore, outer stresses reduce to the pressure applied.
(a)
5
807
35
40
4. Conclusions Fig. 8. X-ray diffractograms of the hcp samples after mechanical grinding for (a) 15 min, and (b) 30 min.
(a)
(b) B
A
A
C
B
B
A
A
Fig. 9. Fragments of hcp (a) and fcc (b) lattices. Arrows indicate the shift of the layers under transformation.
of elasticity still can be used. Within the framework of this theory, the internal energy of hexagonal crystal is determined by Eq. [9] F ¼
k1 k2 ðe11 þ e11 Þ2 þ e233 þ k3 ðe11 þ e22 Þe33 2 2 þ 2k4 ðe213 þ e223 Þ þ ðk1 k5 Þðe212 e11 e22 Þ,
where ki are the elastic moduli. The stress tensor can be found using the relation pij = oF/oeij: 0 1 k3 j 2 0 k4 j B C ð1Þ P¼@ 0 k3 j 2 k4 j A : 2 k4 j k4 j k2 j
Two-layered hexagonal structure of fullerite C60 was found to transform into more stable three-layered cubic structure at shear deformation caused by uniaxial loading and mechanical grinding. The fcc and hcp modifications of C60 changed similarly upon heating in vacuum: the carbon cages of C60 molecule start to break down above 1100 K. No transition was observed after hydrostatic compression. According to the ordinary deformation theory, the hcp–fcc phase transition in single crystals is highly improbable under anisotropic compression (namely, under conditions of our experiments). However, in powders, distribution of stresses on the particles boundaries, which is other than in single crystals, results in shear deformation and the phase transition. This is not the case for experiments with hydrostatic compression: here the particles (even powdered) are dispersed in a liquid; therefore, outer stresses are equal to the pressure applied. Acknowledgement This work was supported by the Russian Foundation for Basic Research, project no. 04-03-32783.
References [1] Heiney P. Structure, dynamics and ordering transition of solid C60. J Phys Chem Solids 1992;53(11):1333–52. [2] Fischer J, Heiney P. Order and disorder in fullerene and fulleride solids. J Phys Chem Solids 1993;54(12):1725–57.
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[3] ArkhngelÕskii I, Skokan E, Velikodnyi Y, Chernyshov V, Sidorov L. Synthesis of a hexagonal close-packed phase of C60 fullerite. Dokl Phys Chem 1998;363(46):413–5. [4] Kurdyumov A, Malogolots V, Novikov N, Pilyankewich A, Shulman L. Polymorphic modifications of carbon and boron hydrides. Moscow: Metallurgiya; 1994. [5] Skokan E, Arkhangelskii I, Zhukova N, Velikodnyi Y, Tamm N, Chelovskaya N. Synthesis of hexagonal modification of C60 using cryoextraction. Carbon 2003;41(7):1387–9.
[6] Skokan E, Privalov V, Arkhangelskii I, Davydov V, Tamm N. Solvent molecules in crystalline C60. J Phys Chem 1999;103(12):2050–3. [7] Tsiklis D. Technique of physicochemical experiments under high and superhigh pressures. Moscow: Khymiya; 1976. [8] Kurdyumov A, Pilyankewich A. Phase transitions in carbon and boron nitride. Kiev: Naukova Dumka; 1979. [9] Landau L, Lifshitz E. Theory of elasticity. 3rd edition. Oxford: Pergamon Press; 1986.
Stability of hexagonal modification of fullerite C60 E.V. Skokan *, I.V. Arkhangelskii, D.E. Izotov, N.V. Chelovskaya, M.M. Nikulin, Yu.A. Velikodnyi Physical Chemistry Department, Moscow State University, VorobÕevy Gory, 119899 Moscow, Russia Received 17 September 2004; accepted 3 November 2004 Available online 8 December 2004
Abstract Stability of the hcp modification of fullerite C60 under heating, compression, and mechanical grinding was studied. It was established that uniaxial load and mechanical grinding caused transition of the hcp modification into more stable fcc modification, while no transition was observed upon heating or hydrostatic pressure. The results were discussed in terms of the theory of the deformation of solids. 2004 Elsevier Ltd. All rights reserved. Keywords: A. Fullerene; B. Arc discharge; C. X-ray diffraction; D. Crystal structure
1. Introduction Fullerite C60 comes in three polymorphic modifications, namely, simple cubic (sc), face-centered cubic (fcc), and hexagonal close-packed (hcp) modifications [1–3]. Cubic and hexagonal close-packed structures can be thought off as being polytypic modifications, since only the packing fashions of identical layers are different in their crystal lattices (sphalerite and wurtzite lattices). The cubic lattices consist of three layers, while the hexagonal ones have two layers. Unlike fcc structures, hcp (wurtzite) structures made of similar atoms are usually unstable under normal conditions and seldom seen in practice [4]. In the case of fullerite C60, the hcp modification is metastable with respect to the stable fcc modification. Therefore, it can be synthesized under undoubtedly non-equilibrium conditions, for example, upon crystallizing from fullerene solutions frozen with liquid nitrogen [5]. Note that cubic modifications of C60 have been studied extensively, while the properties
*
Corresponding author. E-mail address: [email protected] (E.V. Skokan).
0008-6223/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2004.11.007
of the metastable hexagonal modification as an individual compound are poorly understood. The aim of this work was to study the stability of the hcp modification of fullerite C60 under heating, compression, and mechanical grinding.
2. Experimental 2.1. Materials The samples of hcp modification were obtained by extraction from the saturated benzene solution of C60 (fcc) frozen with liquid nitrogen; hexane was used as an extractant [5]. The initial sample of C60 (fcc) contained C60 > 99.98%, C60O and C70 < 0.02%, as established by HPLC. Prior to use, all organic solvents (HPLC purity grade) were distilled. The extracted product was air dried and heated under vacuum at 700 K for 3 h. According to X-ray diffraction date (Fig. 1), its structure was identified as hcp with the following lattice parame˚ , c = 16.38 A ˚ , which are in good agreeters: a = 10.02 A ment with our previous results [3,5].
113
(a)
112
110
222
220
E.V. Skokan et al. / Carbon 43 (2005) 803–808
111
804
10
20
201
114 015 205 222,123 006
121
102
004
101
103
002
100
(b)
30
40
50
60
Fig. 1. X-ray powder diffractograms of (a) fcc and (b) hcp phases of fullerite C60.
The content of the hcp phase in the sample obtained was estimated to be 95% on the basis of the ratio of intensities of the h1 0 3i and h1 1 0i lines taken from the X-ray diffraction pattern. The sample, in which the content of the hcp phase was previously established by the Rietweld method [3], was used as a standard. The amount of residual solvents (0.01–0.02%) was determined from H1 NMR spectra (a Bruker AC-200 spectrometer, room temperature) as in [6]. To identify the solvents, IR spectra of the samples obtained were measured (a Specord M-80 instrument) within the range of 400–4000 cm1; the samples were used in the form of pellets with KBr. No vibrations of the C–H bonds were observed in the IR spectra. 2.2. Apparatus and procedures The material obtained was heated at different temperatures, compressed (uniaxial load and hydrostatic compression) and mechanically ground in a mortar; the content of fcc and hcp modifications after the treatments was monitored at normal conditions (room temperature, p = 1 atm) using X-ray diffraction analysis (a STUDI/P diffractometer, CuKa1 radiation, Ge monochromator, a PSD detector). The reflection with hkl h1 0 3i, which may be observed only for the hcp modification but is forbidden for the fcc, was used for estimating stability of the hcp modifi-
cation, since the change in the intensity of the h1 0 3i line is proportional to the change in the content of the hcp modification in the sample. The description of the line profile was improved by the profile analysis of the X-ray diffraction patterns shown in Fig. 1 that made it possible to consider the reflections absent in the diffraction pattern. All calculations were performed using the WinXPOW program package applied to a STUDI diffractometer. The relative volumes of the samples at different pressures were measured as in [7]. In each experiment, four different weight portions of fcc and hcp samples were used; the pressure changed from 0 to 2.6 GPa with the step of 0.05 GPa. Compression of the samples was studied in a Carver laboratory press using stainless piston-cylinder molds. Mechanical grinding was performed in an agate mortar for 30 min. To study stability of the hcp modification under compression, the sample was pelleted and pellets (30.0 ± 0.5 mg) were subjected to uniaxial loading. A series of experiments was carried out at different pressures (0.3, 0.5, 0.8, 1, 1.6, and 2.7 GPa) but similar time periods (60 min). Another series was performed at similar pressure (0.8 GPa) but different time periods (5, 15, 30, 45, and 60 min). After each run, the load was relieved, and X-ray diffraction analysis of the pellets was carried out. To study the effect of hydrostatic pressure on the hcp sample, the weight portion (ca. 30 mg) was charged into an aluminum ampoule filled with ethanol. The ampoule was shaked for eliminating gas bubbles, then covered with a lid and placed into a press mould. Once the experiment was finished, the product was dried and analyzed by X-ray diffraction method.
3. Results and discussion 3.1. Heating According to X-ray analysis, no phase transition took place upon heating the samples at 750 K for 5 h (Fig. 2). At elevated temperature, the X-ray diffraction patterns showed the decrease in peak intensities with increasing background. At 1200 K, no reflections were observed that is evidence of destruction of the carbon cage of the fullerite molecule. 3.2. Uniaxial load The relative volumes (V/V0) of the fcc and hcp samples vs. load are shown in Fig. 3. The curves obtained for both samples are similar. The X-ray diffraction analysis showed that the fcc samples stayed unchanged after experiments, while the hcp structure transformed into
E.V. Skokan et al. / Carbon 43 (2005) 803–808
805
35
1200 K
30
25
1100 K 20
15
900 K
10
5 0.0
0.5
1.0
1.5
2.0
2.5
3.0
Fig. 4. Intensity of the h1 0 3i peak for hcp sample vs. load.
750 K
5
10
15
20
25
30
35
Fig. 2. X-ray powder diffractograms of the hcp phase after thermal treatment at different temperatures.
fcc structure. The compression curves are smooth and monotonous, thus indicating that the hcp–fcc transition is not discontinuous but goes on within a wide pressure interval; therefore, it can be classified as an martensitetype transition. It should be noted that transition between lonsdelite and diamond, belonging to polymorphic carbon modifications, also proceeds continuously [4].
The extent of transformation of hcp modification into fcc depending on pressure is illustrated in Fig. 4. As seen from this figure, the hcp–fcc transition proceeded within a wide pressure range that is in good agreement with the above data. Note that everywhere over the range studied (up to 2.7 GPa), the hcp modification was not completely transformed into the fcc modification in the timescale of our experiments (Fig. 5). The time dependence of the hcp–fcc transformation at constant load of 0.8 GPa (Fig. 6) can be closely approximated by the equation a 0.07 · t0.45±0.05, where a is the extent of transformation and t is the time. It is rather similar to the time dependence of the martensite-type phase transitions [8]. 3.3. Hydrostatic pressure The sample was compressed at 0.5 GPa for 60 min; however, X-ray diffraction analysis showed no change
1.00
1.00
(b)
(a) 0.98
0.98
0.96
0.96
0.94
0.94
0.92
0.92
0.90
0.90
0.88
0.88 0.0
0.5
1.0
1.5
2.0
2.5
0.0
0.5
1.0
1.5
Fig. 3. The relative volume of (a) fcc and (b) hcp phases vs. load.
2.0
2.5
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E.V. Skokan et al. / Carbon 43 (2005) 803–808
(b)
(c)
(b)
(a)
(a)
5
10
15
20
25
10
Fig. 5. X-ray diffractograms of the hcp samples: (a) initial, (b) subjected to uniaxial load of 0.8 GPa, and (b) 2.7 GPa.
20
30
40
50
60
70
Fig. 7. X-ray diffractograms of the hcp samples: (a) initial, (b) subjected to hydrostatic pressure of 0.8 GPa for 100 min.
3.4. Mechanical grinding
0.5
According to X-ray diffraction data (Fig. 8), grinding results in reducing content of the hcp modification in the sample. Thus, the hcp–fcc transition takes place under mechanical grinding.
0.4
0.3
3.5. Hcp–fcc transformation in fullerite C60 in terms of the deformation theory
0.2
0.1
0.0 0
10
20
30
40
50
60
Fig. 6. The time dependence of the hcp–fcc transformation at constant uniaxial load of 0.8 GPa. The solid line is a least-square fit: a 0.07 · t0.45.
in the sample. The increase in pressure and time period (to 0.8 GPa and 100 min, respectively) virtually did not affect the X-ray diffraction pattern (Fig. 7): only slight change in the ratio of peak intensities was observed. Thus, no hcp–fcc transition took place under hydrostatic pressure.
To understand behaviour of the hcp phase of C60 under mechanical treatment, some aspects of the deformation theory of solids should be engaged. The hcp–fcc transition is accompanied by parallel shift of hexagonal layers (Fig. 9). Let direct the Oz axis across the layers, and the Ox, Oy axes direct symmetrically into the layer plane. When layer A shifts with respect to layer B in the direction shown in Fig. 9, then the strain tensor takes the form 0 1 0 0 j 1B tan a C e ¼ @ 0 0 j A, j ¼ pffiffiffi , 2 2 j j 2j2 where a is the slope angle of the Oz vector. In the case of the hcp–fcc transition, j = 1/2, that corresponds to ductile deformation. However, to evaluate the stress required for initiating this deformation, the linear theory
E.V. Skokan et al. / Carbon 43 (2005) 803–808
Thus, to realize the deformation under study, tangent stresses are required (p13 and p23 do not vanish). Obviously at hydrostatic pressure the stress tensor is spherical over all space, i.e. tangent stresses are absent, and the hcp–fcc transition is impossible. When compression is concentrated along one axis, then
(b)
pij ¼ pli lj ,
10
15
20
25
30
ð2Þ
where l is the unit vector along which the load p is applied. Hence it is evident that anisotropic compression causes non-vanishing tangential stresses. A close look at equations (1) and (2) shows that the stress tensors do not coincide for any l; therefore, in the case of single crystal, the hcp–fcc transition cannot be observed at uniaxial compressive load. In polycrystalline samples, such a coincidence is possible, in principle, since in the latter case, distribution of stresses on the surface and, naturally, in a bulk, is different. However, it is not the fact in the case of hydrostatic pressure. Really, the particles of a sample (even powder) are dispersed in a liquid; therefore, outer stresses reduce to the pressure applied.
(a)
5
807
35
40
4. Conclusions Fig. 8. X-ray diffractograms of the hcp samples after mechanical grinding for (a) 15 min, and (b) 30 min.
(a)
(b) B
A
A
C
B
B
A
A
Fig. 9. Fragments of hcp (a) and fcc (b) lattices. Arrows indicate the shift of the layers under transformation.
of elasticity still can be used. Within the framework of this theory, the internal energy of hexagonal crystal is determined by Eq. [9] F ¼
k1 k2 ðe11 þ e11 Þ2 þ e233 þ k3 ðe11 þ e22 Þe33 2 2 þ 2k4 ðe213 þ e223 Þ þ ðk1 k5 Þðe212 e11 e22 Þ,
where ki are the elastic moduli. The stress tensor can be found using the relation pij = oF/oeij: 0 1 k3 j 2 0 k4 j B C ð1Þ P¼@ 0 k3 j 2 k4 j A : 2 k4 j k4 j k2 j
Two-layered hexagonal structure of fullerite C60 was found to transform into more stable three-layered cubic structure at shear deformation caused by uniaxial loading and mechanical grinding. The fcc and hcp modifications of C60 changed similarly upon heating in vacuum: the carbon cages of C60 molecule start to break down above 1100 K. No transition was observed after hydrostatic compression. According to the ordinary deformation theory, the hcp–fcc phase transition in single crystals is highly improbable under anisotropic compression (namely, under conditions of our experiments). However, in powders, distribution of stresses on the particles boundaries, which is other than in single crystals, results in shear deformation and the phase transition. This is not the case for experiments with hydrostatic compression: here the particles (even powdered) are dispersed in a liquid; therefore, outer stresses are equal to the pressure applied. Acknowledgement This work was supported by the Russian Foundation for Basic Research, project no. 04-03-32783.
References [1] Heiney P. Structure, dynamics and ordering transition of solid C60. J Phys Chem Solids 1992;53(11):1333–52. [2] Fischer J, Heiney P. Order and disorder in fullerene and fulleride solids. J Phys Chem Solids 1993;54(12):1725–57.
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