Calculation of thermodynamic properties for the high-temperature phase of C70 fullerene

__ __ is

s

ELSFIIER

8 September

1997

PHYSICS

LETTERS

A

Physics Letters A 234 (1997) 69-74

Calculation of thermodynamic properties for the high-temperature phase of C,, fullerene J.B.M. Barrio a,b,N.P. Tretiakov

a7c,V.I. Zubov a,c

’ Depnrtamento de Fisica, Uniuersidade Federal de Go&, C.P. 131, 74001.970, Go&k, Go&, Brazil b Deparramento de Matemciticn e Fisica, Uniuersidade Cat6lica de Go&, GoiZnia, Goiris, Brazil ’ Departament of Theoretical Physics, Peoples’ Friendship University, I 17198 Moscow, Russian Federation Received 22 July 1996; revised manuscript received 12 May 1997; accepted for publication Communicated by J. Flouquet

11 June 1997

Abstract An approach of calculating thermodynamic properties of the high-temperature modification of C, is extended to C,O. For the intermolecular forces we use the potential proposed by Verheijen et al. which has been derived by summing Girifalco potentials over the 25 pairs of interacting atomic shells in each pair of molecules. We have calculated the normal isobar of the fee phase which is dominant at high temperatures and its spinodal point. Taking into account available information about the vibrational spectrum of the C,, molecule, thermodynamic properties have been calculated, including

components of the elastic tensor. 0 1997 Elsevier Science B.V.

1. Introduction

Starting with the description of fullerenes by Kroto et al. [I], the advent of efficient synthesis procedures for macroscopic quantities of fullerites by Kratschmer et al. [2], and the discovery of superconductivity in fullerenes doped by metallic atoms by Hebard et al. [3] and Rosseinky et al. [4], a new family of carbon molecules with a closed-cage structure and nearly spherical appearance, have been attractive to the scientific community as well, perhaps because of the prospects for future applications. The structure and properties of C, have been intensively studied recently. In contrast to this, the structure and dynamics of the second most abundant fullerene present in the soluble extract, C,,, have not been fully elucidated. In the fullerenes family, C,, is the next stable molecule following C,,. At low temperatures X-ray diffraction and electron microscopy studies indicate for the lattice structure of C,, the long-range orientational order. At intermediate temperatures there is a mixture of the phases [5-91. At high temperatures the solid C,, has a fee crystal structure with strong anharmonicity of lattice vibrations. In the fee phase the C,, molecules rotate almost: freely around their centre of mass, and present five spherical shells with an equal centre, but different radii and number of carbon atoms [lo- 121. Temperature-dependent structural studies for C,, [5-131 reveal that we cannot specify the transition temperatures with accuracy, because of the existence of metastable phases. The energy difference between the hexagonal and the cubic structure is very small, and a detailed discussion of the thermodynamic stability is 037%9601/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PII SO375-9601(97)00462-3

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J.B.M. Barrio et al./ Physics Letters A 234 (1997) 69-74

difficult. The phase transition from rhombohedral to cubic lattice was estimated in the temperature range from 270 to 340 K, however at high temperatures (T > 360 K) the C,, molecules are orientationally disordered with face-centred cubic structure. The high-temperature structure of C,, has been determined with high-resolution X-ray powder diffraction and electron microscopy, and the structures of the phases are described by a model of complete orientational disorder, consistent with the molecular geometry [I]. There are also discrepancies in the literature for the same phase transitions measured by different groups using different techniques. To describe the transition, some authors [ 14,151 sugested a strong translation rotation (T-R) coupling in solid C,,, responsible for the phase transition. This T-R coupling should also influence the temperature dependence of the elastic constants. In order to confirm the possible influence of orientational-dependent properties on the description, new experimental results are necessary. However, the model of freely rotating molecules described above can be considered as a first approximation, especially at high temperatures ( > 1000 K). The purpose of the present work consists in calculating the complete set of thermodynamic and elastic properties of C,, in such an approximation. We study thermodynamic properties of the fee phase of strongly anharmonic solid fullerene C,a applying the correlative method of unsymmetrized self-consistent field (CUSF) developed by Zubov et al. [ 16-191. This method is efficient for calculations of properties of various crystals: inert gas solids [18], alkali halide crystals [20], some metals [21] and C,, fullerite [22,23].

2. Intermolecular The interaction

forces potential

obtained by Verheijen

et al. [lo] is 1

1

F(R)= -

(ri+rj+R)3

-

(ri-rj+R)’

-

(-ri+rj+R)3

1 (ri+rj+R)9

-

(ri-rj+R)9

1 + (-ri-‘i+R)’

1

1 -

(-ri+rj+R)9

+ (-ri-rj+R)9

1 1

’

(1) where R is the distance between the centres of two C,, molecules, ri the radius of shell i, Ni the number of carbon atoms in shell i, and A and B constants. The values for these parameters appear in Table 1, and the potential function presents a minimum at r. and the depth of the well is c/k. Three sets of parameters are Table 1 Geometrical parameters of a C,, molecule and parameters of the potential (l), as predicted by Pan [I 11, measured by Baker [12] (electron difraction), and potential parameters as measured by Vaughan [s], the depth of the well (s/k) and the minimum point (ra) Multiplicity N, 10 20 20 10 10 A (eV A61 B (eV A’*) c/k (K) r” (A)

R; (A) Pan 1111 3.565 3.665 3.876 4.029 4.172 16.136 15324 3441 10.551

R, (A.) Baker [12] 3.424 3.690 3.875 3.985 4.097 17.153 119993 3436 10.552

Ri (A) Vaughan [5] 3.565 3.663 3.876 4.029 4.173 18.558 25205 3430 10.619

J.B.M. Barrio et al./Physics

Letters A 234 (1997) 69-74

71

available: (i) the theoretical shape of C,, molecule as predicted by Pan et al. [ 111, (ii) the shape determined electron diffraction [12] and (iii) as measured by Vaughan et al. [5].

by

3. Method In the CUSF method [ 16-191, the anharmonic terms up to the fourth order are included in the zeroth approximation and higher terms are taken into account using perturbation theory. The equation of state for the hydrostatic pressure is [ 191 + (3-p)@

dK,

4K,

da

1’

Here @ = kT, u(a) is the volume of the unit cell, a is the nearest-neighbour distance, molecule in the static lattice, K, and K, are the force coefficients and p( K,/3K,/O) the equation B(x)

‘3XD

(2)

iK,( a> is the energy per is a function defined by

D-,,A X + 5P/6X) -~.s(x+WW

’

(3)

where D, are parabolic cylinder functions. When the interactions are pairwise and central, as it is in the case of C,, in fee phase, the force coefficients are given by K,,=

&

C&(r) n#O

; r=

(4)

A-,

here I=O, 1,2 ,..., A^ is the lattice matrix and n are unit vectors with integer components. The cohesive energy per mole is [17] U,=

-(N,/2)[K,+#+P)@],

(5)

where NA is the Avogadro number. The equation of state is of the form P = P, + P, + P,, where P, is the zeroth-order approximation (2), P, and P, are corrections of the perturbation theory [18], and the higher-order anharmonicity [19] respectively. These corrections have the same general structure as the zeroth-order terms, but are more cumbersome. We have calculated the derivatives of the interaction potential (1) and the force coefficients (4) for different sets of parameters (Table 1). These coefficients may be further inserted in the equation of state, on the assumption that pressure is constant. The equation for the nearest-neighbour distance versus temperature was thus derived. It was solved numerically and the thermodynamic properties of the C,, crystal were obtained. To take into account the intramolecular degrees of freedon, we have used the vibrational frequencies calculated by Onida et al. [24] and Jishi et al. [25] and applied the method of calculation of specific heats described in Ref. [23]. Based on the self-consistent theory of elastic properties of strongly anharmonic crystals [26], we have calculated the components of elastic constant tensors and the sound velocities for the C,, crystal.

4. Results and discussion The solution of the equation of state (2) provides spinodal temperatures T, equal to 1916, 1941 and 1949 K for the three sets of parameters respectively: theoretical, experimental and of Vaughan. For temperatures higher

J.B.M. Barrio et al./ Physics Letters A 234 (1997) 69-74

72

Table 2 Calculated properties of the fee phase of C,,,: nearest-neighour distance a, thermal expansion coefficient (Y (10m5 Km ‘1, density p (g/cm’), cohesive energy I/ (J/mole), specific heats C, , C,, (J/mole K), elastic properties B,, B,, CcrD, C:, , a,,, = C,, - Cr2.s (MPa), - C, z ), sound velocities V,, V, (m/s). Results obtained using different parameters of the potehtial: 1 Pan [ 111; 2 anisotropy A = 2C,/(C,, Baker [12] and 3 Vaughan [5] T(K)

a(A) P a -U cv C,,

1 2 3

1 2 1 2 1 2 1 2

1 2

B,

C;, c/z c 44 C II s

1 2 1 2 1 2 1 2 1 2

1 2

c,‘z 6,. 6, A

1 2 1 2 1 2 I 2

V,

’

V,

2 1 2

300

500

700

900

1100

1300

1500

1700

1900

10.570 10.570 10.637 1.672 1.672 0.934 0.955 178826 178960 544.78 547.82 549.75 549.75 16538 15499 16597 15553 26100 24410 11760 11040 13300 12440 26160 24470 11810 11090 1547 1403 1488 1349 1.85 1.86 3955 3825 2820 2728

10.590 10.591 10.659 1.662 1.662 1.007 1.028 176092 176230 986.64 986.72 990.03 989.99 14539 13657 14588 13703 23950 22430 9832 9272 12230 11450 24000 22470 9882 9317 2396 2178 2347 2133 1.73

10.613 10.614 10.682 1.652 1.651 1.101 1.122 173183 173329 1255.67 1255.7 1 1260.57 1260.53 12641 11904 12690 11950 21870 20500 8027 7606 11120 10430 21920 20550 8076 765 1 3098 2825 3048 2780 1.61 1.62 3642 3527 2595 2513

10.637 10.639 10.707 I.640 1.639 1.222 1.243 170047 170206 1410.60 1410.68 1417.34 1417.25 10825 10227 10877 10275 19820 18610 6329 6037 9963 9365 19670 18660 6381 6084 3634 3329 3582 328 1 1.48 1.49 3480 3373 2484 2390

10.665 10.667 IO.736 1.621 1.626 1.389 1.408 166623 166803 1503.19 1503.28 1512.15 1512.02 9059 8595 9114 8645 17750 16700 4716 4542 8711 8224 17800 16750 4770 4592 3995 3682 3941 3632 1.34 1.35 3307 3209 2313 2248

10.697 10.699 10.769 1.613 1.612 1.633 1.647 162793 163015 1561.54 1561.63 1573.47 1573.22 7321 6988 7377 7040 15610 14740 3177 3113 7326 6969 15660 14790 3233 3165 4149 3856 4094 3804 1.18 1.20 3116 3029 2131 2079

10.736 10.738 10.809 1.595 1.594 2.036 2.034 158343 158636 1600.14 1600.22 1616.55 1616.05 5559 5364 5616 5416 133 10 12640 1685 1726 5733 5537 13360 12690 1742 1779 4048 3811 3991 3758 0.98 1.01 2894 2821 1895 1863

10.788 10.790 10.860 1.573 1.572 2.904 2.829 152617 153224 1626.68 1626.80 1651.96 1650.58 3661 3629 3718 3682 10630 10230 176 328 3759 3791 10680 10280 233 382 3582 3462 3525 3409 0.72 0.76 2606 2557 1546 1553

10.888 10.878 10.947 1.530 1.534 10.71 7.086 143081 144710 1645.22 1645.47 1742.76 1708.26 904 1333 957 1384 6322 6722 - 1808 1363 469 1137 6376 6773 - 1754 - 1312 2277 2500 2223 2449 0.11 0.28 2041 2101 553.7 860.9

1.74 3800 3677 2712 2624

than T,, the equation of state has no real roots, and T, is for C,, slightly greater than calculated for C,,. In Fig. 1 we show the nearest-neighbours distance versus temperature for the first two cases. The lower branch of the curves represents stable thermodynamic states and the upper branch stays for unstable ones. Compared with C 60, the values of the nearest-neighbour distance for C,, are somewhat higher. On the other hand, the real branches of the curves in Fig. 1 are nearly equal, except for the Vaughan curve (not presented here) which is somewhat elevated. The fit to available experimental data is quite good. As in the case of C, [22], it is possible to estimate the melting temperature of C,,, T,, = 1500 K.

J.B.M. Barrio et al./ Physics Letters A 234 (1997) 69-74

Fig. I. Normal isobar of the fee phase of C,,, calculated neighbour distance in angstrom.)

with theoretical

(1) and experimental

73

(2) parameters

of the potential.

(Nearest

The complete set of thermodynamic properties of the fee modification of C,, fullerite is listed in Table 2. We present here the values corresponding to the first two cases (theoretical and experimental) and the frequencies of internal vibrations calculated by Onida et al. Calculations using the frequencies calculated by Jishi et al. provide very close results. This is rather surprising considering that the values of the frequencies differ essentially [24,25]. Since experimental data for thermal properties of the high-temperature C,, phase are inadequate to perform comparison, it remains to compare with the properties of C,, [22,23] calculated elsewhere. The behaviour of all functions (isothermal bulk modulus, specific heats, thermal expansion coefficient, elastic constants) are in perfect analogy to those of C,. The specific heats of C,, are somewhat higher and the expansion coefficient somewhat below the corresponding values of C,,.

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