Heat of sublimation
show rapid molecular reorientation in the high temperature phase with little anisotropy [7.108,109]. Some authors have reported that as the temperature is lowered, the fee structure is continuously transformed by deformation into a rhombohedral structure with the long diagonal threefold axis aligned parallel to the (111) direction of the fee structure (see Fig. 7.13) [7.96]. These authors further report at yet lower temperatures, but still above T0l, a stable hexagonal phase (space group P63/mmc) with lattice constants a — b = 10.56 A and c = 17.18 A and a nearly ideal c/a ratio of 1.63 (see Fig. 7.13). The observed crystal structures above 337 K tend to be strongly influenced by crystal quality, thermal history, amount of supercooling, and the amount of C60 impurity in the crystal [7.97]. Differences in transition temperatures at r01 are observed on heating and cooling. It is believed that the most reliable and reproducible results regarding phase transitions are obtained upon warming [7.21]. Some authors believe that the fee phase is the lowest equilibrium energy phase from high temperatures down to Tm = 345 K and do not report phase transitions above T0l [7.21], Differential scanning calorimetry studies on very high purity C70 solids show a sharp feature at 361 K and a second smaller feature at 343 K, associated with the onset of quasi-isotropic tumbling above Tm [7.97],
In the intermediate range between ~280 K and ~340 K, anisotropy in the crystal structure becomes established, as the long axis of the C70 rugby balls begin to align, and the C70 molecules tumble about rhombohedral axes. Many groups see evidence for a first-order structural phase transition near Tm ~340 K, below which anisotropy begins to appear. The phase transition near r01 has been studied by NMR [7.108-111], fiSR [7.112], neutron scattering [7.113], x-ray diffraction [7.95,114,115], dilatometry [7.115], and thermal conductivity [7.81]. The value of the transition temperature r01 has been shown to be sensitive to the thermal history and the amount of solvent in the sample. Some authors identify the intermediate temperature phase with rhombohedral symmetry (space group R3m), which is derived from the high temperature fee structure by the elongation of the fee structure along a unique (111) direction and is driven by the freezing out of the molecular rotation about an axis perpendicular to the long axis [7.8]. Some C70 samples show a transition to another hep lattice below ~337 K, but with a = b = 10.11 A and a c/a ratio of 1.82, larger than that above T01 [7.96]. This larger c/a ratio is associated with the orientation of the C70 molecules along their long axis, as the free molecular rotation (full rotational symmetry), that is prevalent in the higher temperature phase above Tou freezes into rotations about only the fivefold axis of the C,Q molecule (see Fig. 7.13) [7.96]. In this intermediate temperature phase (which encompasses room temperature), all research groups report crystal anisotropy. It is also observed that a ratcheting motion around the long axis of the O;0 molecules is dominant, with the anisotropy of the molecular reorientation increasing with decreasing temperature [7.108-110]. The molecular orien-tational ordering of the C70 molecules in this phase is confirmed by ¿¿SR experiments [7.112],
As the temperature is further lowered to T02 «280 K, the free rotation about the c-axis also becomes frozen. The crystal phase below T02 is a monoclinic structure (see Fig. 7.13) with the unique axis along the c-axis of the intermediate temperature hep or rhombohedral structures, and the monoclinic angle ¡3 is close to 120° [7.96]. Differential scanning calorimetry studies on very pure C70 crystals show TQ2 = 282 K and another small feature at 298 K has been reported [7.97]. The total heat of transformation from 270 K to 380 K is 10.4 J/g. The supercooling effect for the phase transition near T02 is large (with a width of 15-50 K) while the supercooling effect near the T0l transition is much smaller (2-3 K) [7.97]. The difference between the experimental values for T01 and T02 from one research group to another may be due to large supercooling effects, the presence of minority C70 phases, and to the presence of C50 and solvent impurities (which suppress Tm). The molar volume decrease at both the Tm and TQ2 phase transitions of C70 is ~3%, which is considerably lower than the cor responding decrease in solid C60 at Tm. Evidence for molecular alignment is also provided by the qualitative change in the intermolecular pair correlation function that is found between 200 K and room temperature [7.116].
The space group for the low-temperature phase has not yet been clearly identified, with Cm, C2, P2, and Pm being the most likely candidates, and special preference is currently being give to C2 and P2 [7.21]. It has also been suggested that the low-temperature phase may be orthorhombic Pbcm or Pbnm rather than monoclinic, based on a model for the low-temperature orientational ordering of solid C70 [7.117]. Detailed x-ray studies indicate that the low-temperature phase of a sample grown by sublimation as an hep crystal above Tm is of the orthohexagonal Pbnm space group [7.107].
One suggestion for the stacking arrangement of the C70 molecules in the monoclinic phase is shown in Fig. 7.14, where it is noted that one of the
Fig. 7.14. One model for the stacking of orientationally ordered C70 molecules in close-packed planes in the monoclinic phase. A and B represent the two different orientations of molecules within a close-packed layer. The molecules A' and B' are positioned in the adjacent layer. Electron-rich and electron-poor regions are indicated by + and respectively [7.96],
Fig. 7.14. One model for the stacking of orientationally ordered C70 molecules in close-packed planes in the monoclinic phase. A and B represent the two different orientations of molecules within a close-packed layer. The molecules A' and B' are positioned in the adjacent layer. Electron-rich and electron-poor regions are indicated by + and respectively [7.96], former hexagonal a-axes of the hep phase is doubled [7.96]. The former hexagonal plane in the low-temperature phase consists of alternating rows of fixed C70 molecules which are oriented with their main symmetry axes along the unique axis of the monoclinic structure and the cross sections are arranged within rows as shown in Fig. 7.14, so that the high electron density hexagon-hexagon edges are opposite low electron density hexagonal face centers [see Fig. 7.7(b)]. The stacking perpendicular to this plane can be arranged in either of two ways: one where the A and B rows of the C70 molecules are directly over one another, and the other where the C70 molecules in the next layer are shifted to locations bm/2 + cm/2, in which the monoclinic cm axis is related to the hexagonal ah axis by cm — 2ah (see Fig. 7.14). Transmission electron microscopy (TEM) diffraction patterns for both stacking arrangements have been reported [7.86].
The molecular reorientation times rr along the long axis drop rapidly as T is lowered. Some typical values are Tr — 5 ps at 340 K, ir = 2 ns at 250 K, and rr ~ 20 ¿is at 100 K. Also reported are reorientations of the molecular axes at a temperature near Tm, with values of Tr ~ 20 /us reported for rotation of the long axis of the molecule above 300 K [7.108110]. Ratcheting motions of the molecules along the long axis have been reported even at low T (down to 73 K [7.118]).
Because of the small sample sizes currently available for the higher fullerenes C„c, structural studies by conventional methods (such as x-ray diffraction and neutron scattering) are difficult. Therefore, only a few-single crystal structural reports have, thus far, been published regarding the higher fullerenes. The most direct structural measurements thus far have come from scanning tunneling microscopy (STM) studies [7.119,120] and selected area electron diffraction studies using electron energy loss spectroscopy (EELS) [7.121],
The STM measurements have been done on C76, Cg0, C82, and C84 fullerenes adsorbed on Si (100) 2 x 1 and GaAs (110) surfaces [7.119,120] (see §17.9.5). The STM studies provide detailed information on the local environment of the higher-mass fullerenes in the crystal lattice. These STM studies show that all of these higher fullerenes crystallize in an fee structure with C„ - C„ nearest-neighbor distances of 11.3 A, 11.0 A, 11.74 A, 12.1 A for C76, C78, C82, and C84, respectively, and fee lattice constants are obtained from these distances by multiplication by -<¡2. Since higher fullerenes generally have multiple isomers (see §3.2), the fullerenes on the fee lattice sites are not expected to be identical, and therefore more lattice disorder is expected for higher-mass fullerene crystals. Also the more massive fullerenes are less mobile rotationally, so that the rotational ordering phase transition at r01 is driven to higher temperatures, if a phase transition occurs at all.
Fig. 7.15. Electron diffraction profile of a thin film of C76 recorded in an EELS spectrometer with a momentum transfer resolution of 0.06 A [7.121], The inset shows the lattice constant for the highermass fullerenes C„c as a function of y/n^, where 60 < nc < 84.
As mentioned above, electron diffraction measurements provide detailed results on lattice constants and grain sizes of very small crystalline samples of the higher fullerenes. Such measurements were made on thin films of higher fullerenes prepare by sublimation of a purified powder onto an NaCl substrate. The NaCl substrate was then dissolved to yield a free-standing fullerene film with grains of ~1000 A in size [7.121], A trace showing the diffracted beam intensities from elastic scattering within an electron energy loss apparatus is shown in Fig. 7.15 for C76. By indexing the diffraction peaks observed for C60, C70, C76, and Cg4 to an fee lattice, the lattice constants afcc shown in the inset to Fig. 7.15 are obtained and plotted as a function of y/n^, where nc is the number of carbon atoms in C„c. Results for the lattice constant are in good agreement with the STM results given above and in §17.9.5.
The effect of pressure on the structure of fullerenes has been studied for crystalline C60 and C70. Because of the weak van der Waals forces between the fullerene molecules in the lattice, C60 and C70 are expected to be highly compressible at low pressures where the intermolecular distance is reduced, but nearly incompressible at high pressures because of the very low compressibility of the molecules themselves [7.122]. Experiments show, however, that crystalline C60 and C7(l undergo a phase transition before the incompressibility effect is expected to become important (at ~50 GPa) [7.122].
Experimental studies of the pressure dependence of the structures of crystalline C60 and C70 have focused on both the pressure dependence of
MOMENTUM TRANSFER (A-1)
MOMENTUM TRANSFER (A-1)
the structural parameters for the stable phases at atmospheric pressure and the structures of the pressure-induced phases.
As stated above, solid C60 under pressure is rather compressible. Accurate measurements on high-quality single-crystal samples yield values for the bulk modulus /3 = —Vdp/dV of 6.8 GPa for the fee phase above T0l and 8.8 GPa for the sc phase below Tm (see Table 7.1) [7.5,6], in contrast to early reports of values for /3 between 14 and 18 GPa [7.4, 24,25]. These lower values of ¡3 imply a uniaxial compressibility of 2.3 x 10"12 cm2/dyn, which is about the same as the interlayer compressibility [-(1 /c0)(dcQ/dp)] of graphite 2.98 xlO"12 cm2/dyn [7.123], Regarding the structure of solid C60, pressures of up to 15 GPa (keeping the temperature constant) have little effect on the crystal structure, except that the transition temperature T01 increases approximately linearly with pressure (see Fig. 7.16) at a rate of ~105—164 K/GPa depending on the pressure environment [7.5,9,10] (see §7.1.3). Model calculations for the intermolecular C60-C60 interactions yield a value of dT01/dp = 115 K/GPa [7.85,90], in good agreement with bulk modulus experiments and also differential scanning calorimetry experiments (163 K/GPa) [7.5]. At room temperature, the phase transition starts at 135 MPa, but the transition is smeared out over the pressure range 135-500 MPa. The phase transition appears to be first order only in the limit p ->• 0. A linear pressure dependence
Fig. 7.16. Temperature-pressure phase diagram for C60 crystals (dot-dash line). The triangle at 298 K indicates the approximate end of the transition. The onset of the transition (circles) and the maximum slope \dV/dp\ (squares) show a linear fit to the data points (solid lines) [7.5].
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