J . Phys. Chem. 1992, 96, 7150-7153
7150
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on this method are now in progress. Acknowledgment. We express our thanks to Lh.H. Shinohara and Dr. Y.Saito of Mi'e University for providing us the Cso fullerene.
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References and Notes (1) Gordon, J. P.; Cleite, R. C.; Moore, R. S.;Porto, S.P. S.;Whinnery, J. R. J . Appl. Phys. 1965,36,3. Kogelnik, H. Appl. Opr. 1965.4, 1562. Hu,
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wave l e n g t h / nm Figure 4. Change of the real part of the refractive index due to the photoexcitation to the triplet state (An) (a) and the imaginary part (Ak) of the triplet state (b) and the ground state (c) of Cmin benzene. The vertical scale for Ak is expanded by a factor of 4.
large contribution of PL in the signal indicates that the probe beam variation under the TL experimental condition should not be analyzed only by the TL effect, in particular in a short-time region. If photoexcitation induces photoreaction and the lifetime of the product is on the order of milliseconds, the PL contribution in a signal obtained even with a CW laser excitation might be significant. The time constant of the TL signal is determined by a transit time of acoustic wave ( T ~ ~which ) , is defined by T~~ = w/c (w is the radius of the excitation beam; c is the speed of sound in the medium). Under a usual experimental condition, the time constant is on the order of submicroseconds. It means that the dynamics of a short-lived excited state below submicrdgeconds cannot be investigated by the TL method. On the other hand, since the time constant of creation of PL is determined by the creation of the excited state, there is no inherent time limitation in the PL method. Therefore, this PL method can be an alternative useful method to investigate the dynamia of the excited state. Experimentsbased
C.; Whinnery, J. R. Appl. Opr. 1973, 12, 72. (2) Fang, H. L.; Swofford, R. L. In Ultrusenririue Laser Spectroscopy Kliger, D. S.,Ed.; Academic Press: New York, 1983 and references therein. (3) Long, M. E.; Swofford, R. L.; Albrecht, A. C. Science 1976,191, 183. Jansen, K. L.; Harris, J. M. Anal. Chem. 1985,57, 1698. Bialkowski, S. E.; He, Z . F. Anal. Chem. 1988,60,2674. Franko, M.; Tran, C. D. Anal. Chem. 1988,60, 1925. (4) Bailey, R. T.; Cruickshank, F. R.;Pugh, D.; Guthrie, R.; Johnstone, W.; Mayer, J.; Middleton, K. J. Chem. Phys. 1982, 77, 3453. Siebert, D. R.; Grabiner, F. R.; Flynn, G. W. J. Chem. Phys. 1984,60,1564. Toselli, B. M.; Walunas, T. L.; Barker, J. R. J . Chem. Phys. 1990, 92, 4793. (5) Rossbroich, G.; Garcia, N. A.; Braslavsky, S.E. J . Phorochem. 1985, 31, 37. (6) Terazima, M.; Azumi, T. Chem. Phys. Lerf. 1987,141,237; 1988,145, 286; 1988,153,27. Terazima, M.; Kanno, H.; Azumi, T. Chem. Phys. Left. 1990, 173, 327. (7) Fuke, K.; Ueda, M.; Itoh, M. J. Am. Chem. Soc. 1983, 105, 1091. Terazima, M.; Tonooka, M.; Azumi, T. Phorochem. Photobiol. 1991,54, 59. ( 8 ) Poston, P. E.; Harris, J. M. J . Am. Chem. Soc. 1990, 112, 644. (9) Longhurst, R. S.In Geomerricul und Physicul optics, 2nd ed.;Wiley: New York, 1967. Nelson, K. A.; Casalegno, R.; Miller, R. J. D.; Fayer, M. D. J. Chem. Phys. 1982, 77, 1144. (10) Herman, M. ,%;Goodman, J. L. J. Am. Chem. Soc. 1989, I l l , 1849, 9105. (11) Arbogast, J. W.; Darmanyan, A. P.; Foote. C. S.;Rubin, Y.; Diederich, F. N.; Alvarez, M. M.; Anz, S.J.; Whetten, R. J. J . Phys. Chem. 1991, 95, 11. (12) Terazima, M.; Hirota, N.; Shinohara, H.; Saito, Y. J . Phys. Chem. 1991, 95, 9081. (13) Kajii, Y.; Nakagawa, T.; Suzuki, S.;Achiba, Y.; Obi, K.; Shibuya, K. Chem. Phys. Le??.1991, 181, 253. (14) Hare, J. P.; Kroto, H. W.; Taylor, R. Chem. Phys. Len. 1991, 177, 394. ( I S ) Berthoud, T.; Delorme, N.; Mauchien, P. Anal. Chem. 198!3,57,1216. (16) Twarowski, A. J.; Kliger, D. S.Chem. Phys. 1977, 20, 253.
Determination of the Radial Distribution Function of SmaiCParticie Polymer Latices Using Reverse Monte Carlo Simulation Cergely Tbtb* and LAszlb Pusztai Laboratory of Theoretical Chemistry, L. E a t v k University, Budapest 1 1 2., P.O.B. 32., H-1518, Hungary (Received: April 23, 1992: In Final Form: June 23, 1992)
The radial distribution function g(r) has bccn calculated for a dilute polymer latex containing particles with an average radius of 25.6 nm. For this purpose the resulting structure factor, S(K), of a light-scattering measurement was used as input data for a reverse Monte Carlo (RMC) calculation. In this manner the problematic direct Fourier transformation of S(K) leading normally to g(r) could be avoided. Using the three-dimensional particle configurations that had been given by RMC, some features of the local structure, such as cosine distribution of bond angles, were also calculated.
Introduction It has ken a long-standing problem to obtain reliable real-space information on the microscopic structure of colloid systems. As a first step, the radial distribution function' g(r) would provide insight into the structure of ordered regions at any volume fraction that would be easier to interpret than raw diffraction data given in the inverse space. There are severe practical obstacles to obtaining this real-space information. The main difficulty is due to the rather limited range of the scattering vector, K,in either hght-scattering2 or small-angle neutron-s~attcring~ measurements.
This makes the accurate direct Fourier transform of the experimentally obtained structure factor S(K) difficult and often means that the resulting g(r) contains large truncation errors. Computer simulation technique^'.^ are in principle able to provide g(r), together with more detailed structural information through the Cartesian coordinates of particles. The lack of accurate interparticle interaction potentialsmeans that these methods are not widely applicable to real systems yet. Reverse Monte Carlo (RMC) simulation: a relatively new method for structural modeling, was shown to be effective in
0022-365419212096-7150S03,0010 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 7151
Letters determining characteristics of the local structure6 of a number of atomic and ionic systems. This technique can work on the basii of invme space information such as S(K). It seemed quite natural to use RMC for modeling colloid structures using the available light-scattering data,' for example. RMC usually reproduces diffraction data within experimental uncertainties: so the three-dimensional particle configurations that are present at the end of the simulation can be usad for further structural analysis (see, e.g., ref 6). Polymer latices provide a good example of colloidal systems with a reasonably narrow size distribution of particles7(is., thost of small polydispersity). This property is important in both the traditional and RMC evaluation of any diffraction experiment. For the present study a system of small particles (r = 25.6 nm) at a low volume fraction value (Xv = was s e h d . 7 The light scattering data was available for a relatively wide K range (see Figure 5c of ref 7) and this also encouraged our study.
Theoretical Methods The RMC Technique, The basic reverse Monte Carlo methodology has been described in detail elsewhere;ss6v8therefore, only a short summary is given here. The aim is to mow particles around in such a way that the final three-dimensional particle configuration (is., the final set of Cartesian coordinates) should have a radial distribution function whose Fourier transform gives the best agreement available with experimental (diffraction) S(K). For this purpose an algorithm based on the standard (Metropolis) Monte Carlo method4has been developed: (i) Start with an initial configuration of N points (i.e., particles) in a cube of sides L. Calculate initial g(r) and its Fourier transform, S(K). Also calculate x2,the difference between experimental and simulated S(K)functions. (ii) Select one particle and move it by a random dmplacement. Calculate the new g&), S,(K) and X: for the new position. .: If X: is less than x2, the new (iii) Compare x2 and x amfigmtion is accepted; othcrwb it is accepted with a probability proportional to exp[-(x2 - x:)]. (iv) Repeat the procedure from ii until x2 is converged. Having achieved an agreement that is within the assumed experimental errors, a number of configurations can be sampled and averages over particle positions can be calculated. RMC requires no input potential, but reliable diffraction data are essential. There are several advantages of RMC over traditional evaluation methods, of which the most important is that there is always a physically possible particle arrangement corresponding to an RMC solution. This is not true for the direct Fourier transform of S(K), particularly not at limited K range. ChrrctaiptkaofPuticleCollfiOllntioaseeyondg(r).As it is evident from the previous subsection, RMC creates a threedimensional particle distribution that is consistent with the given diffraction data. Using this set of coordinates; it is possible to obtain a picture of the microscopic (or 'local") structure that is more detailed than the "average structure" given by the radial distribution function, g(r). There are several established tools for analyzing thrtadimensional particle configurations,6 of which two will be mentioned here. A neighbor distribution function, C(m),gives the probability of finding an m-fold coordinated particle, where an "m-fold cuordinated particle" is one with m neighbors within a specified distance r,. r, usually corresponds to the first minimum of g(r) if the fvet coordination shell is to be studied. The more familiar coordination number, 4,is then given by the weighted average CmC(m) = n, m
and C(m) provides a measure of the range of different coordinations in a disordered system. Vectors joining a particular particle to all of its neighbors within r, can be called 'bonds". (These are purely geometrical constructions, and no chemical bonding is implied.) Angles confined by two bonds within the same coordination shell can then be called
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npUn 1. Experimental (solid line) and RMC simulated (dashes) structure factor of the polymer latex.
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Figure 2. Radial distribution function of the polymer latex. The integrated number of neighbors is also shown (dotted curve).
'bond angles". The distribution of cosines of bond angles,
B(cos e), is defined as the number of bond angles with cosines in the interval c08 e to c08 e + ~ ( ~ e). 0 8~ ( ~ e) 0 is s appropriately normalizcdP (For multicomponent systems there may be many BabT(cos8) distributions.) Simulation Details. Consistent with the experimental condit i o n ~N , ~= 4096 spherical particles with diameters of up = 51.2 nm were confined in a cubic box of sides L = 6.608 pm. This configuration gives the experimental number density of p = 14.195 pm-3 which corresponds to a packing fraction value of 7 = 0.001, An fcc lattice served as a convenient starting point for the simulation run. With br = 10 nm as the maximum change allowed in any coordinate, an acceptance ratio of 1/3 was achieved, and 2 million accepted steps were completed. The very low volume fraction accounts for the high acceptance. Throughout the present study it is assumed that the experimental structure factor represents scattering from point sources. According to ref 7, the small polydispersity of the polymer latex has been ignored, and the data have been treated as thost obtained from a monodisperse system. Both approximations work well at low volume fraction values, but they should be investigated in mort detail when considering denser colloidal systems.
Results and MscuMion Experimental and RMC simulated structure factors, S(K), are compared by F i i 1. The level of agreement shown was reached within half a million accepted steps. Figure 2 shows the radial distribution function, g(r), for the polymer latex. This smooth curve is an average taken over five independent particle configurations sampled between 1.2 million and 2 million accepted moves. The stability of g(r) in the quoted regionwas excellent; statistical fluctuatiolls that occurred remained
Letters
7152 The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 10
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F i i 3. Neighbor distribution function calculated up to 400 nm (after 1, 1 million; 2, 1.2 million; 3, 1.4 million; 4, 1.6 million; 5, 1.8 million; 6, 2 million accepted steps).
Figure 4. Cosine distribution of bond angles calculated up to 400 nm (after 1, 1 million; 2, 1.2 million; 3, 1.4 million; 4, 1.6 million; 5, 1.8 million; 6, 2 million accepted steps).
within 1%. It is interesting to note that despite the extremely low packing fraction the radial distribution function has at least three clearly distinguishable peaks, of which the first one is sharp even compared with those found in ionic systems.6 This in itself indicates that there is a strong ordering in radial direction that is far better defined than in hard sphere systems at much higher density.’ The first peak of g(r) at r = 60 nm is obviously caused by two (or more) particles sticking together, near the “hard sphere” diameter (a = 51.2 nm). The second, third, and also the hardly noticeable fourth peaks follow the first one equidistantly, separated by Ar 450 nm, which represents a fairly regular behavior. The small peak Occurring near the first minimum of g(r),between 200 and 400 nm, is probably due to the rarer, but existing, “clusters” consisting of four or more atoms (see Figure 3, neighbor distribution function). IR Figure 2 the integrated number of neighbors as a function of r is also given. From that it is clear that the average number of particles within the “first coordination shell” (i.e., strictly within the first minimum of g(r)) is less than 1, so the high first maximum does not automatically mean that every particle is coordinated to another one. The neighbor distribution function, C(m),is shown in Figure 3, calculated up to 400 nm (Le., up to the end of the plateau of the first minimum of g(r)). The distribution is relatively sharp showing the considerable radial ordering which exists. The measure of angular order up to the same distance can be seen in Figure 4. Angles that correspond to the peak at cos 8 = 1 fall into the region of angles between 8 = Oo and 8 = 15O, according to the spacing of the B(cos e) histogram, which is Acos 8 = 0.05. These rather small angles can be originated to particle triads where two particles are in contact and the third (which is to be considered as the origin of the angle) is further away. The peak in B(cos 8) at cos 8 = 1 therefore supports the idea that in this low volume fraction latex particles tend to form associations of two or even more particles. Taking into accowlt the low volume fraction again, it is not very likely that particles close enough to these associations are to be found so that triads forming acute angles could be set up. This can explain the asymmetric behavior of B(cos e) around
data ~et.~*lO It has also been demonstrated that starting from radically different initial configurations, e.g., from randomly distributed points and from an fcc lattice, gives the same final particle configurations, within statistical fluctuations, even at relatively high volume fraction (q = 0.3)? For extremely dilute systems, however, one would exped another difficulty. As particles have considerable freedom to move around, in long time periods they might realize several local structures that would be different only in terms of B ( m 8) but would give rise to the same g(r). In the present study a survey of this problem was carried out by running the RMC far longer than would have been necessary and investigating neighbor and cosine distributions at time intervals. Results of this test are shown by Figures 3 and 4. As c2n be seen, differences between structures found at different stages of the run are comparable or even smaller than fluctuations observed within individual curves. From this we can conclude that, at least as far as the quantities such as g(r), C(m),and B(cos 8) are concemed, no basically different local arrangements were found that agreed with the experimental data.
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At the low volume fraction region, where coordination spheres are well-separated relative to the particle size and the first coordination sphere does not have a well-defined environment, these methods for describing local symmetries are not that powerful as in the case of more dense materials, such as molten salts. Nevertheless, colloids of much higher packing fraction are well-known,’ and for these systems the distributions described in the previous section would be more relevant for characterizing short-range structures. At such a low density the question of the uniqueness of a given particle arrangement from RMC is important. It has been shown previously that for simple liquids with moderate packing fractions various RMC runs do not give totally different results from a given
Conclusions It should be noted that results given by RMC were shown to be the least ambiguous when the packing fraction was above q = 0.3.” This is understandable considering that the number of available microscopic particle arrangements is smaller if excluded volume effects are more apparent. The applicability of RMC for systems with extremely high packing fraction ( q = 0.5) has also been demonstrated.12 Therefore it is suggested that RMC could be applied successfully for colloids from the high volume fraction region. Reverse Monte Carlo method was shown to be able to derive radial distribution function for polymer latices on the basis of light scattering measurements. On the grounds of the present study we propose that RMC can be used for modeling results of small-angle neutron-scatteringexperiments done on colloid systems of much higher volume fraction. Neighbor distribution and cosine distribution functions were found to be appropriate to indicate structural changes, even if they are not the best tools for describing the actual local structure at particularly low volume fractions. Acknowledgment. We are grateful to Dr. T.Gililnyi for useful discussions. Financial support for this work was provided by the MHB Magyar Tudodnyi5rt (for Hungarian Science) Foundation. Calculations of this scale were made possible by a generous allocation of computer time by the IBM Hungarian Academic Initiative.
References and Notes (1) Hansen, J.-P.; McDonald, I. R. Theory of Simple Liquids;2nd ed.; Academic: London, 1986. (2) Brown, J. C.; hsey, P.N.; Goodwin, J. W.;Ottewill, R.H.J. Phys. A: Gen. Phys. 1975,8,664.
J. Phys. Chem. 1992,96,7153-7156 (3) Jacrot, B. Rep. Progr. Phys. 1976,39, 911. (4) Allen, M. P.; Tildesley, D. J. Compurer Simulation of Liquids;
7153
( 8 ) McGreevy, R. L.; Howe, M. A.; Keen, D. A.; Clauscn, K. N. IOP
Con/. Ser. 1990, 107, 165.
Clarendon Press: Oxford, 1987. (5) McGreevy, R. L.; Pusztai, L. Mol. Simul. 1988, 1, 369. (6) McGreevy, R. L.;Pusztai, L. Proc. R . Soc. London 1990,430, 241. (7) Ottewill, R. H. Progr. Colloid Polym. Sci. 1980, 67, 71.
(9) hsztai, L.; Teth, G. J. Chem. Phys. 1991, 94, 3042. (IO) McGreevy, R. L.; Howe, M. A. Phys. Chem. Liq. 1991,21, 1. (11) T6th, G.; Pusztai, L. Chem. Phys. 1992, 160, 405. (12) Pusztai, L. 2.Naturforsch. 1991, 46a, 69.
An EXAFS Study of the Metallofullerene YC,,:
I s the Yttrium Inside the Cage?
L. Soderbolm,* P. Wurz,K.R. Lykke, D. H. Parker2 Argonne National Laboratory, Argonne, Illinois 60439
and F. W. Lytle The Boeing Company, Seattle, Washington 98124 (Received: May 12, 1992; In Final Form: July 18, 1992)
A sample, determined by time-of-flight mass spectroscopy (TOFMS) to consist of YCs2as the major metal-fullerene complex, was analyzed by X-ray absorption s ectroscopy. The Y is found to have 7 f 1 near-neighbor C atoms at 2.35 0.02 A and an Y neighbor at 4.05 f 0.05 The unequivocal observation of an Y-Y interaction is unexpected, since the mass
* 1. spectral data show no indication of Y2C, as a major component of the sample. We believe that the combined TOFMS and
extended X-ray absorption fine structure (EXAFS) results are not consistent with models that place the metal ion inside the fullerene cage. Instead, we propose that our data can be explained with a dimer of the form Cs2Y-X-YCsz, where -Xis a bridging carbon or oxygen species. The short Y-C near-neighbor distance indicates a strong, bonding interaction between the metal ion and the fullerene cage.
Iatroduction It has been suggested that the cavity created inside the "socarball-like" structure of the C, fullerene clusters could trap metal ions to form endohedral complexes with interesting electron properties. Evidence for the formation of Laca, made by laser vaporization of a low-density graphite source impregnated with LaC13, was obtained by time-of-flight mass spectroscopy (TOFMS).' TOF peaks attributable to Laca or Lac,+ were identified, where n ranged from 44 to greater than 76. There was no evidence of clusters associated with more than one La ion; therefore, it was inferred that there is only one uniquely-stable binding site per C, and that this site is inside the C, cage. Photodissociation of metal-carbon clusters involves the successive loss of C2 fragments, in a pattern similar to that obtained from photodissociation of the bare cluster^.^ The loss of C2 rather than M+has been used to argue that the metal ion is sterically trapped inside the fullerene cage. This suggestion gains further support from the apparent stability of these metal complex= to reaction with either oxygen or moist air.3 In particular, Lacs2 is found to be the only lanthanum fullerene to be produced by standard carbon-arc techniques that expose the soot to air. A comparable stability of LacBz and the bare cluster Cg4has led to the description of the complex in terms of a La2+ion mside a C8f cage? This description has been shown to be inconsistent with EPR' and xpSsdata, both of which indicate that La is nearly trivalent. The latter interpretation is more consistent with the known chemical stability of trivalent La, but the Cs2>counterion is difficult to reconcile with the known stabilities of the bare clusters. Recently, the unique stability of the single-metal-ion clusters has been brought into question by the discovery of a series of complexes LanCsCh (n = 0, 1, 2), with La2Cs, as a major, extractable species from a Laz03-loadedcarbon-arc experiment.6 It has been suggested that the metal atom is incorporated into the cage framework itself, each La ion replacing two carbon atoms, rather than being trapped inside the cage structure.6 'Permanent address: Institute for Surface and Interface Science, Univenity of California, h i n t , CA 92717.
Recently, yttrium (Y) has been reported to be incorporated into the fullerene cages with even greater facility than LaS5 The metallofullerenes YCm and Y2C, are reported, with Y2CB2appearing to be the most abundant dimetal compound. Whether the metal ions are trapped inside or are outside the fullerene cage, or are incorporated into the cage structure itself, is difficult to determine because of both the small sample sizes and the variety of C, impurities present in the samples. We have chosen to use X-ray absorption spectroscopy (XAS) to provide direct information about the environment of the Y ion in these samples. XAS is a particularly suitable technique to study this problem because it is a single ion probe, and therefore by tuning the energy to the absorption edge of interest, it is possible to determine electronic and environmental information about a specific ion in a complex matrix. Experiments Sample R q w a h ud CbuPcterlPtioa.The metallofullerene
samples were prepared in our carbon plasma generat~r.~A 4.3-mm-diameter hole was drilled into a carbon rod of 6.3" thickncas at a length of 38 mm. This rod was filled with a mixture of Y203and graphite of equal volume quantities. The mixture was pressed into the tube, but no binder was used. This rod was inserted in the fullerene generator, carefully pumped down to 1W2 Torr, and burned under our usual experimental conditions for fullerene prod~ction.~ The soot obtained was collected with no special precautions to prevent air expoewe. The soot was sonicated in toluene and centrifuged. The solution was decanted from the residue and dried. A mass spectrum of the material obtained by this procedure is shown in Figure la. The synthesis yielded the well-known distribution of all-carbon molecules, the fullerenes C, C7* C72, ...,Cznup to very high masses (a3000 amu). Other peaks are observed in the mass spectrum between the fullerene molecules, which are identified as series of YCh and Y2Cb molecules. The identified YC2, molecules are YCm, YCu, and larger YCz, entities, with 2n up to 100 and more. The most abundant molecules of this series are YC, YCm YC76,and YCs2 at m / z = 929, 977, 1001, and 1073 amu, respectively. The smallest molecule observed of the Y2C2, series is Y2c74, and
0022-3654/92/2096-7 153$03.00/0 0 1992 American Chemical Society
