The Journal of
Physical Chemistry VOLUME 98, NUMBER 27, JULY 7,1994
Q Copyright 1994 by the American Chemical Society
LETTERS Cluster Origin of Fullerene Solubility Valery N. Bezmelnitsin, Alexander V. Eletskii,' and Eugene V. Stepanov Institute for Applied Chemical Physics, Russian Science Center "Kurchatov Institute", Kurchatov Square, Moscow 123182. Russia Received: February 25, 1994; In Final Form: May 1 I , 1994'
The possibility of the existence of fullerenes in organic solvents in the form of clusters containing a number of fullerene molecules is discussed. A theory is developed on the basis of a droplet model for clusters, which enables one to describe the distribution function of clusters by size and to obtain the non-monotone temperature dependency of fullerene solubility observed recently. It is shown that a phase transition taking place in a crystalline Cbo a t Tc N 260 K significantly changes the role of clusters in solution. At T > Tc the distribution function of clusters, characterized by the number n of constituent molecules, has its maximum a t n* = 11 and higher, while a t T < Tc the value of n* H 3-5. This is the reason for the different temperature dependencies of fullerene solubility in various temperature ranges. Comparison of the calculated values of solubility with experimental data permits one to obtain energetic parameters characterizing the interaction of a fullerene molecule with its surrounding in a solid phase or a solution.
The keen interest in fullerenes,lJ a recently discovered allotropic form of carbon, is due to much of the unusual physical and chemical properties as well as the diversity of their behavior at various conditi0ns.~5 In particular, the experiments performed previously revealed a number of interesting peculiarities characterizing the features of fullerene solutions in organic solvents. So, the solubility of Cm in a number of organic solvents6displays a non-monotone temperature dependence, reaching its peak magnitudeat temperature e 2 8 0 K . AlsoIinref7,itwasobserved that the optical spectra of C ~dissolved O in the mixture of toluene and acetonitrile undergo a sharp transformationas the acetonitrile concentration increases above 60%. This phenomenon was interpreted in terms of the generation of clusters composed of a certain number of molecules of C70. Besides, the results of osmometric measurements* show that the average number of dissolved particles in a Cm solution in chlorobenzeneat T = 343 K is nearly 1.3 times less than the full number of dissolved molecules Cm. All these facts (see also ref 9)indicate a possibility for fullerenes to generate clusters in solutions. As can be readily *Abstract published in Advance ACS Abstracts, June 15, 1994.
0022-3654/94/2098-6665$04.50/0
appreciated, such a possibility can significantly reflect on the properties of those solutions. It is shown in the present work that the non-monotone temperature dependence of the solubility of reported in ref 6 can be explained assuming a cluster nature of the solubility of fullerenes. Thereby, the descent of solubility with temperature at T > 280 K is caused by mere thermal decomposition of the fullerepe clusters. Let us consider the determination of the temperature dependence of the fullerene solubility allowing for cluster formation. We will take into account the experimental data6 exhibiting a decrease in the value of solubility, which is about an order of magnitude in the region of 280 < T < 400 K. This permits us to suppose that, at temperatures around a solubility maximum, the average number of fullerene molecules per cluster is much greater than unity and to use a droplet model for the description of the cluster properties. According to this model, the transition from solid to solution does not change a molecule's individuality. Thus, the thermodynamic energy of a cluster in solution consists of two terms: first, a volume term being proportional to the number n of molecules in a cluster, and second, a surface term determined by the surface square proportional to n2I3. It is in line with the 0 1994 American Chemical Society
Letters
6666 The Journal of Physical Chemistry, Vol. 98, No. 27, 1994
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i
n
6-
3
e
7-
L(
2 6-
i
8 5-
U
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: 40 4
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Figure 1. Solubility of Cm in hexane (+, multiplied by 5 5 ) , toluene ( 0 , multiplied by 1.4),andCS~(*).~ Thesolidcurverepresentsourcalculated data.
above assumption that a cluster, being spherical, contains a large number of molecules (n >> 1). Using a standard thermodynamic approach,lOJ1we represent the distribution functions of clusters in solution over the number n of the constituent fullerene molecules in the following form:
f,,= g,, exp[(-An
+ Bn2’3)/Tl
Here, the parameter A refers to the difference in thermodynamic specific energies of the interaction of a fullerene molecule with its surroundings in the bulk solid phase and within a cluster, and the parameter B is a similar difference in energies for molecules placed on the cluster’s surface. In general, these parameters can be of any sign, but for the distribution function to be normalized the condition A > 0 should be fulfilled. The statistical weight of a cluster g,, also generally depends on n as well as the temperature; however, we neglect those in comparison with the strong exponential dependency (1). This is in line with the assumption n >> 1 for a droplet model. The net solubility of a fullerene is determined by the sum
which, with regard for n
>> 1, can be replaced by integration:
Here, in accordance with the above assumption, the averaged statistical weight g of clusters is taken out of integration. This coefficient comprising an entropy factor can generally take different values for various solvents. The parameters A and B, which determine the temperature dependence of solubility, vary with thenatureofsolvents, too. However, the reported6similarity of the temperature dependencies of relative solubilities for C6o in hexane, toluene, and CS2 implies the closeness of the above parameters for all those solvents. Let us try to use expression 3 for the description of the observed temperature dependencies of Cm solubility presented in Figure 1 in relative units. It is easy to understand that the non-monotone dependence C( T ) can take place only if B > 0 and A > 0. In this case, the integral (3) as a function of temperature can have only a minimum. Indeed, the second derivative of C( T) in the point of extremum is an integral of a positive expression, which is apparently positive. Hence, expression 3 has no maximum in
terms of temperature. However, as mentioned in ref 6, change in the character of the temperature dependence from ascending to a descending one can be caused by a phase transition in a solid Cm occurring at a critical temperature T = 260 K. Owing to this transition a sc lattice is replaced by a fcc lattice. Besides, at those temperatures the inherent rotation of fullerene molecules is unfrozen.4 Experimental investigations of this phenomenon show that the transition refers to a first-order phase transition, is endothermic, and is characterized by specific enthalpy AH = 7.1 kJ/mol.12+13 This fact implies a considerably different role of the cluster at temperatures above and below the critical one. Indeed, the values of the parameter A are not the same for both crystalline structures, but they are connected by the relation
where indices sc and fcc refer to the respective lattices and temperatureregionsbelow theabove T,. It means that thecluster distributions function (1) descends with the number n more strongly at temperatures T < Tcthan at T > Tc. The maximum of this function corresponds to values n 1, so that the contribution of clusters in the net solubility can be considered as negligible. This explainsan ascendingcharacter of thedependence C(T) at T < T,. It should be noted that usually the phase transition involved occurs in a temperature region of a finite width 6T mainly determined by a degree of purification and a method for preparation of a solid fullerene.12.13 Typical values of this parameter are 6T = 1-10 K, though even wider transition regions were reported in several works (see, for example, ref 4). Taking into account the phase transition and eq 4, expression 3 takes the following form
-
c = CJn x
- [ A , + (AH- T6S)B(T,- T ) ] exP(Bn2” T which is applicable throughout the temperature range involved. Here A S is the specific entropy of the transition, and B(T - T,) is a step function. As is shown by numerical computing, expression 5 qualitatively describes the data obtained in ref 6 and presented in Figure 1. The best accord is reached at the following values of the parameters: B = 970 K, Af, = 320 K, CO= 5 X lW[Cm mole fraction]. The dependence of C( T ) computed at these values of the parameters is shown by a solid curve in Figure 1. The calculated dependence proved to be virtually insensitive to the width 6T of the phase transition if 6T < 10 K. Besides, we have tracked how a characteristic size n* of clusters corresponding to the maximum of the integrated expression in ( 5 ) is changed with temperature at the obtained values of the parameters. At T < Tc this size monotonically increases from n*(190 K) = 3 to the value n*(260 K) = 11, which remains practically the same up to T = 380 K. Thereby, at the temperature region T > T,, the value n >> 1, which proves the feasibility of a droplet model, justifies the above assumption that the main contribution to the fullerene solubility at temperatures within the region of descending C( T ) is caused by clusters of large sizes. It is also interesting to note that the obtained characteristic size of clusters is close to the first magic number (n = 13) for fullerene C a clusters observed in the vapor p h a ~ e . 1 This ~ permits us to suggest the similarity of structures of clusters of Cm in both vapor and solution. Thus, the conjecture that the solubility of fullerenes is of a cluster nature, being indirectly suggested by the results of the experiments in refs 7 and 8, enabled us to explain the nonmonotonetemperature dependency of the solubilityof C a observed experimentally for several solvents. In fact, the decrease of solubility on warming is caused by the thermal dissociation of
Letters clusters containing the greatest number of fullerene molecules. Further increase of temperature should ultimately lead to a rise of solubility; however, numerical computations show that the minimum of solubility at the above values of the parameters A and B is expected at T H 4000 K wherein neither a solvent nor a fullerene exists. At low temperatures T C 260 K, the role of clusters is likely to be insignificant, which is connected with the change of a crystalline structure as well as with the increase of the energy of the interaction of fullerene molecules with their surroundings in a solid phase. In closing, it should be noted that, for a cluster nature of the fullerene solubility to be established completely, direct experimental exploration is necessary. As a possible scheme for such experiments (see also ref 15), the measurements of infrared absorptionspectra of a fullerenesolution,involving concentrations at various temperatures and a constant optical path length, can be conceived. The existence of such a dependency will indicate the presence of fullereneclusters in solution. Another way, having been partially realized in ref 8, is based on applying Raoult’s law. According to this law, the saturation vapor pressure of a solvent above a solution differs from that above a pure solvent by a value being proportional to the concentration of solute particles. The measurement of the flow of the solvent vapor, being determined by the difference of the pressures, will enable one to obtain the dependence of the solute particle concentration on the concentration of the solute. Provided this dependence is nonlinear, it will form a basis to draw a quantitative conclusion concerning the existence of the fullerene clusters in solution. At present, the mentioned design of experiments is under development.
The Journal of Physical Chemistry, Vol. 98, No. 27, 1994 6667 Along with solubility, the cluster formation also has to affect the transport properties of fullerenes in solution, particularly diffusion. It can be shown that a characteristic size of clusters decreasesas the fullereneconcentrationdrops below the saturation value. Since the mobility of a cluster increases with its decreasing size, it leads to the dependenceof an effectivediffusioncoefficient on concentration. This can serve as a basis for the method of diffusion separation of higher fullerenes being usually of a tiny concentration in solution. Acknowledgment. The authors are grateful to B. M.Smirnov for helpful advice. This work was partially supported by the Russian Foundation of Fundamental Studies. References and Notes (1) Kroto, H. W.; et al. Nature 1985, 318, 162. (2) Smalley, R. E.;et al. The Sciences 1991,31,22. (3) Kroto, H. W.; Allaf, A. W.; Balm, S. P. Chem. Rev. 1991,91,1213. (4) Eletskii, A. V.; Smimov, B. M. Sou. Phys.-Usp. (Engl. Trawl.) 1993,163,(2),33. (5) Huffman, D. R. Phys. Today, Nov 1991, 23. (6) Ruoff, R.S.;et al. Nature 1993,361, 140. (7) Sun,Y.-P.; Bunker, C. E. Nature 1993,365,398. (8) Honeychuck, R.V.; Cruger, T. W.; Milliken, J. J . Am. Chem. Soc. 1993,115, 3034. (9) Wang, Y.-M.; Kamat, P. V.; Patterson, L. K. J . Phys. Chem. 1993, 97,8793. (10) Landau, L.D.;Lifshitz, E. M. Statistical Physics; Pergamon Press: Oxford. 1977. (11) Smimov,B.M.Sov.Phys.-Usp.(Engl. Trawl.) 1993,162(1),119; 162 (12),97;163 (lo), 29. (12) Heiney, P. A.; et al. Phys. Rev. B 1992,45, 4544. (13) Samara, G.A.; et al. Phys. Rev. B 1993,47,4756. (14) Martin. T.P.: et al. Phvs. Rev. Lett. 1993. 70. 3079. (isj B ~ U W. , J.; i t al. Phyi Rev. Lett. i991,67,i493.
