tries then were refined at B3LYP/6-311+G**. Corre- lated ab initio methods (CCSD( T)/6-311+G** for 1 to 3, MP2/6-31G* and MP2/6-311+G** for 1 to 6) were employed to reoptimize some of the geometries as further checks. Frequency calculations at MP2/6- 311+G** for 1 to 3 and MP2/6-31G* for 1 to 6 further confirm they are minima [Web fig. 2 (3)]. All calculations were carried out using Gaussian G98 (M. J. Frisch et al., Gaussian, Pittsburgh, PA, 1998). 5. Typical reference bond length at B3LYP/6-311+G** are: r CC = 1.531 Å in ethane, 1.396 Å in benzene, and 1.329 Å in ethene; r CB = 1.554 Å in CH 3 BH 2 and 1.396 Å in CH 2 BH; and r BB = 1.629 Å in D 2d H 2 BBH 2 . 6. P. v. R. Schleyer et al., J. Am. Chem. Soc. 118, 6317 (1996). An extended bibliography of NICS is given by S. Patchkovskii and W. Thiel [ J. Mol. Model. 6, 67 (2000)]. 7. The data in Table 1 confirm (i) the expected  electron counts ( tot ) from natural bond orbital (NBO) analysis [A. E. Reed, L. A. Curtiss, F. Weinhold, Chem. Rev. 88, 899 (1988)]; (ii) that the  occupancy increments () contributed by A, B, and C are close to two, one, and zero electrons, respectively; (iii) that the NICS(1) values are consistent with the Hu ¨ckel rule [compounds with 4n + 2  electrons have negative NICS(1), whereas those with 4n  electrons have positive values]; and (iv) the H chemical shifts in aromatic compounds are deshielded [Hs  7.0 parts per million ( ppm)], whereas those in antiaromatic compounds are shielded (Hs  7.0 ppm). 8. The WBI is a measure of the bond order based on natural bond orbital analysis. The individual indices for each of the five bonds to the central carbons varies, but the total WBIs to the ppCs for 4, 5, and 6, as well as other hyparenes, 7 and 8 are close to 4 ( Table 1). 9. K. Exner, P. v. R. Schleyer, Science 290, 1937 (2000). 10. Planar tetracoordinate carbon is increasingly well represented. For the latest examples and literature references, see Z.-X. Wang, P. v. R. Schleyer, J. Am. Chem. Soc. 123, 994 (2001); Z.-X. Wang et al., Org. Lett. 3, 9 (2001). 11. SiB 8 (D 8h ) with a planar silicon, as well as B 9 - (D 8h ) and CB 8 (C 2v ) with a boron in the center are minima. 12. G. A. Olah et al., Hypercarbon Chemistry ( John Wiley, & Sons, New York, 1987). 13. NICS(1) describes the NICS value 1 Å above a ring center, where the  ring current effects dominate over the local  contributions [P. v. R. Schleyer et al., J. Am. Chem. Soc. 119, 12669 (1997)]. 14. In the following discussion, the NICS(1) and proton chemical shifts (H), shown in Table 1, are used as criteria to judge aromaticity and antiaromaticity. 15. P. v. R. Schleyer, K. Najafian, Inorg. Chem. 37, 3455 (1998), and references therein. 16. S. Becker, H.-J. Dietze, Int. J. Mass Spectrom. Ion Processes 82, 287 (1988). 17. Computed vibrational spectra and NMR chemical shifts can be obtained from the authors on request. We will gladly cooperate with experimentalists. 18. F. M. Ge et al, Chem. J. Chinese Univ. 17, 1458 (2001). 19. F. M. Ge et al., Chem. J. Chinese Univ. 18, 1838 (2001). 20. J. Nagamatsu et al., Nature 410, 63 (2001). 21. The D 5h CSi 5 2- dianion and its isoelectronic analogs, C 2v CSi 4 P - and C 2v CSi 3 P 2 , are minima with ppCs at B3LYP/6-311+G**//B3LYP/6-311+G** (P. v. R. Schleyer, K. Exner, unpublished data). 22. The closest analogy is the transition metal planar tetracoordinate carbon complex [see S. L. Buchwald et al., J. Am. Chem. Soc. 111, 397 (1989)]. 23. D. Ro ¨ttger, G. Erker, Angew. Chem. Int. Ed. Engl. 36, 812 (1997). 24. W. Siebert, A. Gunale, Chem. Soc. Rev. 28, 367 (1999). 25. For example, replacing two -(CH) 3 - subunits in cyclooctatetraene by two -C 3 B 2 - groping gives a C 8 H 2 B 4 compound with two planar tetracoordinate carbon atoms. 26. We dedicate this paper to G. A. Olah for his contri- butions to hypercarbon chemistry. Z.X.W. thanks R.-Z. Liu and M.-B. Huang for their encouragement. 20 February 2001; accepted 15 May 2001 Predicting the Mesophases of Copolymer-Nanoparticle Composites Russell B. Thompson, 1 Valeriy V. Ginzburg, 1 * Mark W. Matsen, 2 Anna C. Balazs 1 † The interactions between mesophase-forming copolymers and nanoscopic par- ticles can lead to highly organized hybrid materials. The morphology of such composites depends not only on the characteristics of the copolymers, but also on the features of the nanoparticles. To explore this vast parameter space and predict the mesophases of the hybrids, we have developed a mean field theory for mixtures of soft, flexible chains and hard spheres. Applied to diblock- nanoparticle mixtures, the theory predicts ordered phases where particles and diblocks self-assemble into spatially periodic structures. The method can be applied to other copolymer-particle mixtures and can be used to design novel composite architectures. Mixtures of solid nanoparticles and block co- polymers can yield complex, highly ordered composites for next generation catalysts, selec- tive membranes, and photonic band gap mate- rials (1–3). The specific morphology and hence the utility of these materials depends on the copolymer architecture and on such parameters as the size and volume fraction of the particles. Here, we present a method for calculating the morphology and thermodynamic behavior of copolymer-particle mixtures without requiring a priori knowledge of the equilibrium struc- tures. The method combines a self-consistent field theory (SCFT) for polymers and a density functional theory (DFT) for particles. The SCFT has been remarkably successful in de- scribing the thermodynamics of pure polymer systems (4 ), whereas DFTs capture particle ordering and phase behavior in colloidal sys- tems (5, 6 ). Applied to a diblock-particle mix- ture, this technique identifies new self-assem- bled (SA) morphologies, where both particles and polymers spontaneously order into a meso- scopically regular pattern. We thus delineate conditions where the chains drive particles to self-assemble into continuous “nanowires” or “nanosheets.” The method can also be applied to composites involving other copolymer archi- tectures (triblocks, multiblocks, combs, stars) or blends of different polymers. Our model system consists of a mixture of molten AB diblock copolymers and solid spherical particles. All particles have the same radius R. Each diblock consists of N segments, each of a volume  0 -1 . The frac- tion of A segments per chain is denoted by f. The enthalpic interaction between an A seg- ment and a B segment is described by the dimensionless Flory-Huggins parameter,  AB . As a function of ( AB N) and f, a pure diblock melt can form spatially periodic mi- crostructures with lamellar, cylindrical, spherical, or more complicated phases. In SCF theory, many-body interactions between differing segments are replaced by the interaction of each segment with the av- erage field created by the other segments. Here, w A (r) is the value at a point r of the mean field felt by the A segments, w B (r) denotes the field for B segments, and w p (r) represents the field for particles. Using this approach, the free energy (7 ) for our system is given by F = F e + F d + F p (1) The first term, F e , details the enthalpic inter- actions in the system: F e = 1 V  dr [ AB N A (r )  B  r ) +  BP N B  r  p  r  +  AP N A  r  p (r )] (2) where V is the volume of the system,  AP and  BP are the interaction parameters between the respective segments and particles, and  A (r),  B (r), and  p (r) are the dimensionless concentrations of A segments, B segments, and particles, respectively. The diblock en- tropic free energy F d is adapted from (4 ): F d =  1 -  p  ln  V 1 -  p  Q d  - 1 V  dr  w A  r  A  r ) + w B (r  B  r  (3) where Q d is the partition function of a single diblock subject to the fields w A (r) and w B (r). The overall volume fraction of particles is 1 Chemical Engineering Department, University of Pittsburgh, Pittsburgh, PA 15261, USA. 2 Polymer Sci- ence Centre, University of Reading, Whiteknights, Reading RG6 6AF, UK. *Present address: Dow Chemical Company, Building 1702, Midland, MI 48674, USA. †To whom correspondence should be addressed. E- mail: balazs1@engrng.pitt.edu R EPORTS www.sciencemag.org SCIENCE VOL 292 29 JUNE 2001 2469