Size dependences of the lattice parameter and thermal expansion coefficient of C60 fullerite nanoparticles

NanoStruduredMaterials.Vol. 8. No. 5, pp. 595-603.1991 Elsevim Science Ltd CB1991 Acta Metallurgica Inc.

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SIZE DEPENDENCES OF THE LATTICE PARAMETER AND THERMAL EXPANSION COEFFICIENT OF Cm FULLERITE NANOPARTICLES V.I. Zubov*+, I.V. Mamontov+

and J.N. Teixeira Rabelo*

*Universidade Federal de Goi& C.P 13 1,74001-970 Goi%ria - Go, Brazil +Peoples’ Friendship University, 117419, Moscow, Russia (Accepted July 28,1997) Abstract-- Using the statistical theory of structural, dynamic and thermodynamic properties of anharmonic crystalsandsmallparticles, we investigate size effects in Ceofullerite nanoparticles withfcc lattice. The intermolecularpotential derived by Girtfalco is utilized. We have considered three highly symmetricalformsofparticles: cubical, spherical andoctahedral, and calculatedsize dependences of their mean lattice parameters a and thermal expansion coefficients a. Influences of the surface tension andof the lattice relaxation near sulfates have been taken into account. For all threeforms of nanoparticles, both a and aincrease with decreasing size; that is a consequence of the anharmonicity. This effect is appreciably enhanced when the temperature increases. We also note a dominant contribution of the lattice relaxation to the size dependences. 0 1997 Acta Metallurgica Inc. INTRODUCTION

The thermodynamic size effects in nanostructured systems (ultradispersive powders, thin films, especially the granulated systems, etc.) are related to the developed interphase surface, in whose vicinity there is an essential portion of atoms or molecules and that for this reason gives a remarkable contribution to the properties of such systems (l-3). The near-to-surface molecules differ from those that are located in the bulk by a fewer number of neighbors, greater vibration amplitudes and stronger anharmonicity and anisotropy of these vibrations. This gives rise to the following phenomena: lattice relaxation, i.e. the variation of the interplanar distances, and also close to the non-singular faces a shift between the near surface molecular layers, sometimes suqace reconstruction, i.e. a variation of the symmetry of the surface layers, and excess surface thermodynamic functions such as the surface energy, free energy and tension, etc. Recently, a statistical theory of the structural, dynamic and thermodynamic properties of surfaces of anharrnonic crystals was developed on the basis of the correlative unsymmetrized self-consistent field @JO-12), and a discussion on the perspectives of its applications to the study of the thermodynamic size effects in the nanostructured systems can be found in ref. (13,14). In this work we use it for the calculation of the size dependence of the mean lattice parameter and the

596

VI Zueov, IV MAMONTOV AND JN

TEIXEIRA RABELO

coefficient of thermal expansion in fine particles of Cso fuller& with fee lattice that occurs at temperatures above 261.4 K (15). This investigation is interesting because in the production of the fullerenes usually a soot is obtained, i.e. a fine-dispersive powder (1516). Earlier, the CUSF has been applied to the study of bulk thermodynamic properties of this fullerite (17,18) and also of its equilibrium with gaseous phase (19). A good agreement with available experimental data takes place. BASIC FORMULAE The size dependences of the mean lattice parameter and of the thermal expansion coefficient of a tine crystal particle of regular shape are determined by the formulae &/LZ=(2Cq>-~Kr<“P)//;

[ll PI

where 1 = na is the distance between opposite faces, n the number of molecules along an edge, a, a, KT and y are the lattice parameter, the thermal expansion coefficient, the compressibility, and the surface tension coefficient of a homogeneous material respectively, q is the normal to the face component of the displacement of the surface layer and l.t a dimensionless coefficient depending on the shape, in particular, on the number of its faces (shape-function). The brackets c > mean averaging over all faces of the particle. The first (relaxation) terms in the right sides of [l] and [2] show the influence of a local distortion of the lattice near the interphase surface. The second (quasi-Laplacian) ones express a quasi-homogeneous compression or tension of the whole particle under the action of the surface tension, that for a solid can be either positive or negative. Expressions [l] and [2] are the generalization of known formulae for spherical particles to polyhedral ones. In CUSF it is possible to calculate all quantities that appear in [l] and [2] if the intermolecular interactions are known. In the fee modification of the fulleriteC6o moleculesrotatealmost freely. Therefore, the part of the intermolecular forces that depends on the orientation vanishes. Using the Lennard-Jones (12-6) potential for the interactions between the atoms of two Gee molecules and averaging over all their orientations, Girifalco (20) has obtained the following intermolecular potential for the gaseous and fee phases of this fullerene:

&j!&jy-$

[31

where s = r/2a, a = 3.55 x 10e8cm, a = 7.494 x lo-l4 erg, and p = 1.3595 x lo-l6 erg. Its minimum point is r-o= 10.0558 x low8cm, and the depth of potential well is c/k= 3218.5 K. The forces which are described by this potential are more short-ranged than in the simple van der Waals crystals. Besides, their Debye temperature is about 3.5 times lower than the equilibrium point with the orientationally ordered phase. Therefore, the material in question can be characterized as a

SIZE DEPENDENCES AND THERMAL EXPANSION OF C,

FULLERITE NANOPARTICLES

597

classical van der Waals crystal with a great number of intramolecular degrees of freedom. Its intramolecular vibrations are harmonic, while the anharmonicity in the lattice vibrations is very essential. The zerofh order approximation of CUSF (4,5,8) is basedon nonlinear integral equations for the one-particle probability densities of atoms or molecules and their self-consistent potentials. The solutions for these equations include the fundamental contribution of the main anharmonic terms without using perturbation theory. Taking into account terms up to the fourth order, the equations of a homogeneous crystal with cubic symmetry and intramolecular degrees of freedom have the form (5,18) PO dK2 --+p_, 2K2 da

(15 + p>o 2

(3- p)O dK4 4K4

+fix,gocothz ;JJ

da I

- 1

.

[41

[51

Here, 0 = kBT. a is the distance between lattice points, v(a) = V/N is the volume of the unit cell, oj and gj are the frequencies of the intramolecular vibrations and the degree of their degeneracy, Ko (a) is the potential energy of the static lattice per molecule, K2 and K4 are the force coefficients of second and fourth order, and finally p(K2/(30K4)v2 ) is an implicit function determined by the transcendental equation (5)

[61 where&

are the parabolic-cylinder functions. In the case of pair-wise forces, in particular [3], K2,=-

’ CZ,V2’@(R,), 2/+1&l

I= 0,1,2,

where Zk and Rk are the coordination number and radius. The statistical perturbation theory allows one to improve the contribution of the main anharmonic terms and take into account higher order anharmonicity (6,7,9). The crystal described by these equations is the comparison system for the nanostructured material. From [4] and [5] it is seen that the intramolecular vibrations have no influence over the thermal equation of the fuller&e, and therefore have no influence over such properties like thermal expansion and isothermal elasticity. But their contribution to the energy characteristics are very important. For instance, it is about 90-95% for the heat capacity. The size dependence of the internal energy is determined basically by the static term in [5], i.e. by the number of broken bonds in the vicinity of the interphase surface. The specific or molar heat capacity in general does not depend on the degree of dispersion because the surrounding molecules have no influence on the intramolecular vibrations. That is why we do not consider size effects on the thermal properties.

VIZueov,IVMAM~NTOV

598

AND

JN

TEIXEIRA RABELO

We restrict ourselves to the case of weak anharmonicity, i.e. to the case of not so high temperatures

x = &(3 / @&)“2 >> 1, p(X) ^I 3[ 1-+(1+)].

WI

We shall take into account only nearest neighbor interactions, which for the fullerite Cso with fee lattice assures 90% of the cohesive energy. Therefore, in the absence of external forces P + 0 (in the equilibrium with vapor) the solution of the equation of state [4] with the correction by the perturbation theory (9) have the form

a(O)=rg--_Z;

I

[ g+

E+$[k-?$+!$

13g2

5h+‘Q+W

2rof

r-0 J$

r91

Ii$

where ro is the minimum point of the intermolecular potential, and f = @“Q-n), g = @“‘(ru), h = QN(ro),

k = O”(Q).

DOI

5h+%+81f t-0 G$ Z$

1111

Hence the thermal expansion coefficient is determined by

--+%zh 2f

1%T3 13S2 4f2+ 2rof

For the potential of Girifalco [ 1, lo] we have

a = ro[l+0.0340*(l+0.5370*];

o( = Y(l+

1.07460*),

WI

where@*=@/E. From the above mentioned nonlinear integral equations, in the case of a crystal with surfaces, it follows a set of transcendental equations for the mean displacements of the molecules from the sites of the perfect lattice. It determines the lattice relaxation, the dispersion of their positions, which express the effective amplitudes of its anharmonic vibrations, and also the higher order moments (8). Solutions for these equations were obtained for the singular faces of the fee lattice by expanding the quantities that we are seeking in powers of temperature (10,12). In the linear approximation near the faces {00 1) and ( 111) , only one interplane distance varies, while in the quadratic, two. The lattice relaxation near the surfaces { 110) has a monolayer character being non monotonic. For the displacement of the surface layer of the face { 001 } the following expression is obtained

SIZE DEPENDENCES ANDTHERMALEXPANSION OF C,

165h

15448

FULLERITE NANOPARTICLES

5287f

599

[I31

The general formulae for the other surfaces have analogous forms. Instead of giving them here, we write only the numerical results for all the three singular faces of the fullerite Go: q(OO1)= O.O1604reO* (1 + 0.6510*) ; q(ll1) = O.O1515roO* (1 + 13370*); q( 110) = O.O2577r@* (1 + 1.3330*).

[I41

The thermodynamic functions of the surface of an anharmonic crystal in CUSF are obtained in form of power series of temperature and of the components of the surface strain. In the absence of external stresses (in the equilibrium with the vapor), we have for the surface free energy densities and surface tensions of the singular faces of the & fuller&

o(OOl)=~[l-0.3520*(1+0.3120*)]; a

o(llO)-

y(~l)~-4,~3@*/r;;

E& o(lll)---T[1-0.3530*(1+0.6210*)] a

[15]

--&[l-0.3620*(1+0.6550*)]:,

y(lll)=-2.7410*/r;;

y(llO)--3.8980*/&

~[161

Hence, the tensions of all the three singular surfaces are negative. Values of the isothermal bulk moduli BT = KT-*, necessary for the calculation of the dimensional effects [l] and [2], are given in the work (18).

RESULTS AND DISCUSSION We have considered three high symmetric forms of nanoparticles: cubic, octahedral (Figure 1) and spherical. The cubic particle has 6 faces (001}, 12 edges and 8 vertices. It contains N = n(4n2 - 6n + 3)~

UW

Ns = 12(n-1)2 + 2

[17bl

molecules, whose

600

VI Zutw,

IV MAMONTOV ANDJN TEIXEIRARAEELO

lie on the surface, including [17cl

Ne = 12(n - 2)

molecules on the edges and 8 in the vertices. For it in the formulae [l] and [2] = q(OOl), cp = @Ol) and p = 2/3.

The octahedral particle has 8 faces { 111),12 edges and 6 vertices. In this case (18) [181

N = n(2n2 + 1) / 3, Ns = 2(2n2 - 4n + 3), Ne = 12(n - 2), = q(11 l), <‘y>= tilll)

and p = 8/9.

The spherical particle is an idealized model with smoothed edges and vertices. Assuming that its surface consists of singular faces (001 }, ( 111} and { 110) , the area of each one being inversely proportional to the density of its free surface energy [ 151, we have

TABLE 1 The Number of Molecules along the Edge n, the Total Number of Molecules N, their Number at the Surface Ns, on the Edges and in the Vertices Nev, their Fractions, and the Effective Radii (in units of the intermolecular distance a) ref(3N/47ca)t/3 for the Cubic (the first lines) and Octahedral (the second lines) Particles

100 200 500

83350 3940300 666700 3.176E+7 5333400 4.965E+8 8.333E+7

9606 117614 39206 475214 158406 2988014 996006

582 1184 1182 2384 2382 5984 5982

11.5 3.0 5.9 1.5 3.0 0.6 1.2

6.1 1.0 3.0 0.5 1.5 0.2 0.6

24.14 87.29 48.28 175.0 96.56 438.2 241.4

SIZE DEPENDENCESANDTHERMALEXPANSIONOF C,

FULLERKENANOPAATICLES

(4

601

(b)

Figure 1. (a) The cubic fee cluster with n = 4; (b) the octahedral one with n = 5.

cq> ~0.0187~~0*(1+

1.14509;


= - 3.52w

&

1191

and p = 4/3. The maximum value of this shape function for the spherical particle is due to the fact that it is the limiting case of a regular polyhedron when the number of faces tends to infinity. Since for all singular faces 4 > 0, dq/dT> 0,y < 0.and &/dT < 0, the mean intermolecular distance and thermal expansion coefficient of nanostructured fuller&e C, are more than those of its homogeneous crystal. It is seen from [ 171and [ 181and Table 1 that the dependences N(n) and iVs(n)for cubic and octahedral particles differ essentially from one another, and the same is true for size effects [ 11and [2]. Because of this, it is more convenient to express &z/a and 6akx in terms of the total number of molecules N or its effective radius re_ = (3N / 47~a)“~ a, Table 1. (Clearly for the spherical particle, ref= r). Calculations show that with such variables,results for all three shapes of particles are very close to one another. That is why we show size dependence of each value only with one of the shapes: the cubic one for the lattice parameter and octahedral for the thermal expansion coefficient. Figures 2 and 3 demonstrate size effects at 26 1.4 and 1400 K in the range of r,ffrom 4 to 200 nm. The first temperature is the equilibrium point with the orientationally ordered phase. The second one is close to the melting temperature estimated (17,18) at 1400-1500 K. One can see that these effects rise sharply with the temperature. For the lattice parameter it remains within a fraction of percent. However, for the thermal expansion coefficient, a difference from that for a monocrystal amounts to tens of percent in the vicinity of the melting point. Note that the strong anharmonicity which occurs at high temperatures builds up this effect. We can suppose that the

602

VI Zueov, IV MAMONTOVAND JN TEIXEWA RABELO

5E-37

1 -261.4K 2-1400K

0

100 150 200 SO Effective radius (in units of a)

Figure 2. Size effect in the mean lattice for cubic particles.

0

150 200 100 50 Effective radius (in units of a)

Figure 3. Size effect of the thermal expan-

sion coefficient for octahedral particles.

parameter

distinction between experimental values of o!obtained for a pure monocrystal (2 1) and for the soot (22) at least partially is explained by those size effects that we are describing here. The relaxation term gives the dominant contribution to the size effects, except for very high temperatures where the contribution to the thermal expansion coefficient of both terms are close to one another. In conclusion, we note that unlike those properties studied here, the specific heat of a fuller&e practically is not subject to size effect. This is due to the fact that this quantity is 80-90% determined by intramolecular vibrations, which rather does not depend on environment (23) in particular on the proximity of a surface. ACKNOWLEDGMENTS This work was supported in part by Conselho National de Desenvolvimento Cientifico e Tecnologico - CNPq (Brazil). REFERENCES 1. 2. 3. 4.

5. 6.

Morokhov, I.D., Trusov, L.I. and Lapovok, V.N., Physical Phenomena in Ultradisperse Media, Energoatomizdat, Moscow, 1984, in Russian. Siegel, R.W., Annual Review of Materials Science, 1991,21,417. Marks, L.D., Report of Progress in Physics, 1994, 57,603. Zubov, V.I. and Terletsky, Ya.P.,Annals of Physics, 1970, (Leipzig) 24,97. Zubov, V.I., Annals of Physics, 1974 (Leipzig) 3 1, 33. Zubov, V.I., Physica Status Solidi (b), 1975,72, 71,483.

Metallic

SIZE DEPENDENCES ANDTHERMALEXPANSION OF C,

7. 8. 9. 10. 11.

FULLERITE NANOPARTICLES

603

Zubov, V.I., Physica Status Solidi (b), 1978,87,385; 1978,88,43. Zubov, V.I., Physica Status Solidi (b), 1982, 111,417. Zubov, V.I. and Mamontov, I.V., Bulletin Peoples’Friendship University, 1995, No. 3, Issue 1,80, in Russian. Zubov, V.I. and Tretiakov, N.P., Physica Status Solidi (b), 1991, 164,409. Zubov, V.I., Mamontov, I.V. and Tretiakov, N.P., International Journal of Modern Physics, 1992, B6, 197.

12.

13. 14. 15. 16. 17. 18. 19.

Zubov, VI., Tretiakov, N.P. and Teixeira Rabelo, J.N., Applied Surface Science, 1996, 92,436. Zubov, V.I., Nanostructured Materials, 1993,3, 189. Zubov, V.I., Nanostructured Materials, 1995,5,571. The Fullerenes, eds.H.W. Kroto, J.E. Fischer and D.E. Cox. Pergamon, Oxford, 1993. Datars, W.R., Gal@ S., Olech, T. and Ummat, PK., Canadian Journal of Physics, 1995,73,38. Zubov, V.I.,Tretiakov, N.P., Teixeira Rabelo, J.N. and Sanchez Grtiz, J.F., Physics LettersA, 1994, 194,223. Zubov,V.I.,Tretiakov,N.P., Sanchez, J.F. andCaparica,A.A., PhysicalReview, 1996,B53,12080. Zubov,V.I., Sanchez-Ortiz, J.F.,TeixeiraRabelo, J.N. andzubov, I.V., PhysicalReview, 1997,B55, 6747.

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22. 23.

Girifalco, L.A., Journal of Physical Chemistry, 1992,96,858. Heiney, PA., Vaughan, G.B.M., Fischer, J.E., Coustel, N., Cox, D.E., Copley, J.R.D., Neumann, D.A., Kamitakahara, W.A., Creegan, K.M., Cox, D.M., McCauley Jr., J.P. and Smith III, A.B., Physics Review, 1992, B45,4544. Scanlon, J.C. and Ebert, L.B., Journal of Physical Chemistry, 1993,97,7138. Martin, M.C., Du, X., Kwon, J. and Mihaly, L., Physical Review, 1994, B50, 173.