tries then were refined at B3LYP/6-3111G**. Corre-lated ab initio methods (CCSD( T)/6-3111G** for 1to 3, MP2/6-31G* and MP2/6-3111G** for 1 to 6)were employed to reoptimize some of the geometriesas further checks. Frequency calculations at MP2/6-3111G** for 1 to 3 and MP2/6-31G* for 1 to 6further confirm they are minima [Web fig. 2 (3)]. Allcalculations were carried out using Gaussian G98(M. J. Frisch et al., Gaussian, Pittsburgh, PA, 1998).

5. Typical reference bond length at B3LYP/6-3111G**are: rCC 5 1.531 Å in ethane, 1.396 Å in benzene, and1.329 Å in ethene; rCB 5 1.554 Å in CH3BH2 and1.396 Å in CH2BH; and rBB 5 1.629 Å in D2d H2BBH2.

6. P. v. R. Schleyer et al., J. Am. Chem. Soc. 118, 6317(1996). An extended bibliography of NICS is given byS. Patchkovskii and W. Thiel [ J. Mol. Model. 6, 67(2000)].

7. The data in Table 1 confirm (i) the expected pelectron counts (ptot) from natural bond orbital(NBO) analysis [A. E. Reed, L. A. Curtiss, F. Weinhold,Chem. Rev. 88, 899 (1988)]; (ii) that the p occupancyincrements (Dp) contributed by A, B, and C are closeto two, one, and zero electrons, respectively; (iii) thatthe NICS(1) values are consistent with the Huckelrule [compounds with 4n 1 2 p electrons havenegative NICS(1), whereas those with 4n p electronshave positive values]; and (iv) the H chemical shifts inaromatic compounds are deshielded [dHs . 7.0 partsper million (ppm)], whereas those in antiaromaticcompounds are shielded (dHs , 7.0 ppm).

8. The WBI is a measure of the bond order based onnatural bond orbital analysis. The individual indicesfor each of the five bonds to the central carbonsvaries, 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 literaturereferences, 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. SiB8 (D8h) with a planar silicon, as well as B92 (D8h)

and CB8 (C2v) 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 p ring current effects dominateover the local s contributions [P. v. R. Schleyer et al.,J. Am. Chem. Soc. 119, 12669 (1997)].

14. In the following discussion, the NICS(1) and protonchemical shifts (dH), shown in Table 1, are used ascriteria 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. IonProcesses 82, 287 (1988).

17. Computed vibrational spectra and NMR chemicalshifts 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 D5h CSi5

22 dianion and its isoelectronic analogs,C2v CSi4P2 and C2v CSi3P2, are minima with ppCs atB3LYP/6-3111G**//B3LYP/6-3111G** (P. v. R.Schleyer, K. Exner, unpublished data).

22. The closest analogy is the transition metal planartetracoordinate carbon complex [see S. L. Buchwaldet al., J. Am. Chem. Soc. 111, 397 (1989)].

23. D. Rottger, 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 2(CH)32 subunits incyclooctatetraene by two 2C3B22 groping gives aC8H2B4 compound with two planar tetracoordinatecarbon atoms.

26. We dedicate this paper to G. A. Olah for his contri-butions to hypercarbon chemistry. Z.X.W. thanksR.-Z. Liu and M.-B. Huang for their encouragement.

20 February 2001; accepted 15 May 2001

Predicting the Mesophases ofCopolymer-Nanoparticle

CompositesRussell B. Thompson,1 Valeriy V. Ginzburg,1* Mark W. Matsen,2

Anna C. Balazs1†

The interactions between mesophase-forming copolymers and nanoscopic par-ticles can lead to highly organized hybrid materials. The morphology of suchcomposites depends not only on the characteristics of the copolymers, but alsoon the features of the nanoparticles. To explore this vast parameter space andpredict the mesophases of the hybrids, we have developed a mean field theoryfor mixtures of soft, flexible chains and hard spheres. Applied to diblock-nanoparticle mixtures, the theory predicts ordered phases where particles anddiblocks self-assemble into spatially periodic structures. The method can beapplied to other copolymer-particle mixtures and can be used to design novelcomposite architectures.

Mixtures of solid nanoparticles and block co-polymers can yield complex, highly orderedcomposites for next generation catalysts, selec-tive membranes, and photonic band gap mate-rials (1–3). The specific morphology and hencethe utility of these materials depends on thecopolymer architecture and on such parametersas the size and volume fraction of the particles.Here, we present a method for calculating themorphology and thermodynamic behavior ofcopolymer-particle mixtures without requiringa priori knowledge of the equilibrium struc-tures. The method combines a self-consistentfield theory (SCFT) for polymers and a densityfunctional theory (DFT) for particles. TheSCFT has been remarkably successful in de-scribing the thermodynamics of pure polymersystems (4), whereas DFTs capture particleordering 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 particlesand polymers spontaneously order into a meso-scopically regular pattern. We thus delineateconditions where the chains drive particles toself-assemble into continuous “nanowires” or“nanosheets.” The method can also be appliedto composites involving other copolymer archi-tectures (triblocks, multiblocks, combs, stars) orblends of different polymers.

Our model system consists of a mixture ofmolten AB diblock copolymers and solidspherical particles. All particles have thesame radius R. Each diblock consists of Nsegments, each of a volume r0

21. 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 bythe dimensionless Flory-Huggins parameter,xAB. As a function of (xABN) and f, a purediblock melt can form spatially periodic mi-crostructures with lamellar, cylindrical,spherical, or more complicated phases.

In SCF theory, many-body interactionsbetween differing segments are replaced bythe interaction of each segment with the av-erage field created by the other segments.Here, wA(r) is the value at a point r of themean field felt by the A segments, wB(r)denotes the field for B segments, and wp(r)represents the field for particles. Using thisapproach, the free energy (7) for our systemis given by

F 5 Fe 1 Fd 1 Fp (1)

The first term, Fe, details the enthalpic inter-actions in the system:

Fe 51

V E dr[xABNwA(r)wB~r)

1 xBPNwB~r!wp~r! 1 xAPNwA~r!wp(r)] (2)

where V is the volume of the system, xAP andxBP are the interaction parameters betweenthe respective segments and particles, andwA(r), wB(r), and wp(r) are the dimensionlessconcentrations of A segments, B segments,and particles, respectively. The diblock en-tropic free energy Fd is adapted from (4):

Fd 5 ~1 2 fp! lnFV~1 2 fp!

QdG

21

V E dr@wA~r!wA~r) 1 wB(r!wB~r!#

(3)where Qd is the partition function of a singlediblock subject to the fields wA(r) and wB(r).The overall volume fraction of particles is

1Chemical Engineering Department, University ofPittsburgh, Pittsburgh, PA 15261, USA. 2Polymer Sci-ence Centre, University of Reading, Whiteknights,Reading RG6 6AF, UK.

*Present address: Dow Chemical Company, Building1702, Midland, MI 48674, USA.†To whom correspondence should be addressed. E-mail: balazs1@engrng.pitt.edu

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given by fp, and the prefactor 1 – fp is thevolume fraction of the diblocks. Finally, theparticle entropic free energy is given by

Fp 5fp

alnSVfp

QpaD 2

1

V Edrwp~r!rp(r)

11

V Edrrp(r)Chs@w# p~r!# (4)

where Qp is the partition function of a singleparticle subject to the field wp(r). The localparticle volume fraction, wp(r), is related tothe dimensionless center of mass distribution,rp(r), by

wp(r) 54a

3pR3 E r9 ,R

dr*rp~r 1 r9! (5)

The parameter a denotes the particle-to-diblock volume ratio

a 54pR3r0

3N5

4p

3 S R

R0D 3

N# 1/ 2 (6)

where R0 [ aN1/2 is the natural size of thepolymer, a is the statistical segment length(assumed to be the same for both A and Bsegments), and #N [ a6r2N is the invariantpolymerization index. The last term of Fp

describes the excess (non-ideal) steric freeenergy of the particles through the DFT de-rived by Tarazona (5, 8). In particular, Chs isthe Carnahan-Starling function (9) for theexcess free energy of a hard-sphere fluid,evaluated with the “weighted” (locally aver-aged) particle density, w# p(r). This form of theDFT can be readily extended to ellipsoidalparticles (6). Thus, our model can be gener-alized to mixtures of copolymers and rods or

platelets. By using the same method to com-pare spherical and ellipsoidal particles, wecan determine the effects of particle shape onthe self-assembly process and equilibriumstructure of the system.

In the mean-field approximation, wA(r),wB(r), and wp(r) are determined by locatingsaddle points in the free energy functional(Eq. 1) subject to the incompressibilityconstraint

wA~r) 1 wB~r) 1 wp(r) 5 1 (7)

This yields a system of equations that issolved numerically and self-consistently togive possible equilibrium solutions. To ob-tain these solutions, we implement the com-binatorial screening technique of Drolet andFredrickson (10, 11). The advantage of thistechnique is that it requires no a priori knowl-edge of the equilibrium morphology; this isparticularly important in our studies becausethe diblock-particle system can exhibit struc-tures that are as yet unknown. We make aninitial random guess for the fields and calcu-late all the densities and the free energy ateach step; the fields are then recalculated andthe entire process is repeated until changes inthe diblock densities at each step becomesufficiently small. In addition, we also mini-mize our free energy with respect to the sizeof the simulation box, as proposed by Boh-bot-Raviv and Wang (12).

The system is specified by seven param-eters: xABN, xAPN, xBPN, N# , f, R, and fp. Wefix xABN 5 20 because we are only con-cerned that the value be sufficiently large todrive the microphase separation of thediblocks. To model an affinity between theparticles and one of the blocks (A), we treat

the particles as chemically identical to A,setting xAP 5 0 and xBP 5 xBA 5 xAB. Thiscauses the particles to be preferentially local-ized in the A domains. For #N, we assign atypical value of 1000.

The calculations are first performed in onedimension, allowing for composition varia-tions in only one direction. Thus, we canresolve lamellar morphologies and investi-gate the distribution of particles within thesystem. The particle radius is initially set atR 5 0.3R0. The results of varying the particlevolume fraction, fp, at f 5 0.35 are shown inFig. 1, A and B. With fp 5 0.15, Fig. 1Areveals a new SA morphology where theentire composite displays spatial ordering.The A density distribution along the lateraldirection, wA(x), shows a noticeable “crater”in the center of the A lamellae, indicating anexclusion of A chains from this region. Theprofiles for the particles, wp(x), and theircenters, rp(x), show that the spheres are lo-calized within this cavity. The entire systemhas organized into a well-ordered “core-shell” structure, with the particles formingnanosheets within the polymer matrix. Whenthe particle volume fraction is lowered tofp 5 0.03 for the same system (Fig. 1B),there are too few particles to form a solid coreand thereby exclude the A blocks from thecentral region. Thus, the composite cannotform the highly ordered SA phase.

Figure 1C shows a system with smallerparticles (R 5 0.20R0) and a volume fractionof fp 5 0.15. Instead of the SA phase seen inFig. 1A, we now observe a new “edge-assem-bled” morphology, with the particles forcednearer the AB interface (as shown by theparticle center distribution). The A-block

Fig. 1. Concentration profile of diblock-particle systems [wA (x), densitydistribution of A blocks; wp(x) and rp(x), distributions of particles and particlecenters, respectively]. (A) Large particle (R 5 0.3R0), particle volume fractionfp 5 0.15. (B) Large particle (R 5 0.3R0), fp 5 0.03. (C) Small particle

(R 5 0.2R0),fp 5 0.15. B blocks are not depicted. Three periods are shownto better illustrate the morphologies. Sketches showing the structure of eachsystem accompany the profiles.

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concentration is increased in the center of theA domain. Thus, our model indicates thatvariations in the size and volume fraction ofparticles can be exploited to control the par-ticle distribution inside the composite.

These results are in qualitative agreementwith recent experimental studies by Lauter-Pasyuk et al. (13–15) on thin films of sym-metric polystyrene-polybutylmethacrylate (PS-PBMA) diblock copolymers with incorporatednanoparticles. Large nanoparticles (6 nm) local-ize in the center of the lamellar PS domainswhile smaller particles (4 nm) concentrate at thePS-PBMA interfaces, in agreement with Fig. 1,A and C. Our results also agree with recentMonte Carlo simulations and scaling theory byHuh et al. (16) on melts of diblocks and parti-cles in three dimensions; the study predicts thepresence of a SA lamellar structure (Fig. 1A) atcomparable values of R, fp, and f.

The differences among the three structuresin Fig. 1 can be understood by considering therelative entropic contributions of the A blocksand particles. For small volume fractions oflarge particles, the A subchains must stretch toget around the dispersed spheres, incurring aloss in conformational entropy. As fp is in-creased, the stretching of the A segments isreduced by the segregation of the particles intoa central core. (In the SA phase, this free energygain for the polymers offsets the loss of trans-lational entropy for the particles.) On the otherhand, for the same volume fractions of smallparticles, the stretching required by the poly-mers to circumvent the spheres is less signifi-cant. Hence, it is the translational entropy of theparticles that dominates the behavior of thesystem. As more small particles are added tothe system, the spheres are more uniformlydispersed and the entropic free energy per Ablock increases.

Our theory allows us to break the freeenergy into components a posteriori, and totest this hypothesis. In Fig. 2, the A-blockconformational entropic contribution to thefree energy per polymer chain is shown asa function of the particle volume fractionfp. For larger particles, the entropic freeenergy of the A block indeed decreaseswith increasing fp; for smaller particles,this effect disappears.

The previous calculations were performedin one dimension. However, the addition ofparticles can, in principle, cause the formationof more complex structures, requiring a two- oreven three-dimensional description. To checkthe stability of the lamellar morphologies andconfirm the formation of particle nanosheets,we performed a two-dimensional study of thesystem described in Fig. 1C (17). The resultsshow an increase in the central A-block con-centration, as expected from the one-dimen-sional calculations for the edge-assembled la-mellar phase (Fig. 3, A and B), as well as theeffects of crystallization within the A domain as

Fig. 2. A-block entropicfree energy contribu-tion –TSA (where SA isthe conformational en-tropy of the A block)per polymer chain, forlarge particles (R 50.3R0, red curve) andsmall particles (R 50.15R0, black curve), asa function of fp.

Fig. 3. Representations of a diblock-particle system (R 5 0.20R0, fp 5 0.20) in two dimensions.Density plots and surface plots are given for the A-block distribution (A and B), the particledistribution (C and D), and the particle center distribution (E and F).

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the particles localize to fixed positions (Fig. 3,C to F). This cannot be observed in the one-dimensional case. Two-dimensional calcula-tions for the large-particle case are not depicted,but they also confirm the morphology shown inone dimension by Fig. 1A. Additionally, crys-tallization effects are seen in the two-dimen-sional calculation, just as in the small-particlecase. We note, however, that the detailed crystalarrangement may change in three dimensions.

The method introduced here allows us tocapture the synergistic interactions that occurin self-assembling mixtures of flexible copol-ymers and hard particles. It reveals how en-thalpic and entropic effects in multiphase me-dia contribute to the observed morphologiesand, as in our case, can lead to entropicallydriven particle crystallization. The versatilityof this theory allows us to study the role ofcopolymer architecture, particle shape, andcomposite composition, all without making apriori assumptions about the structure of thesystem. Furthermore, we can compare thefree energies of competing structures andconstruct equilibrium phase diagrams. Over-all, the technique can offer valuable assist-ance in exploring the rich variety of meso-structures that can occur in polymer-particlemixtures and in designing hybrid materialswith the desired morphology.

References and Notes1. M. Templin et al., Science 278, 1795 (1997).2. D. Zhao et al., Science 279, 548 (1998).3. S. Stupp, P. V. Braun, Science 277, 1242 (1997).4. M. W. Matsen, F. S. Bates, Macromolecules 29, 1091

(1996) and references therein.5. P. Tarazona, Mol. Phys. 52, 81 (1984).6. G. J. Vroege, H. N. W. Lekkerkerker, Rep. Prog. Phys.

55, 1241 (1992) and references therein.7. The “free energy” discussed here is actually a

dimensionless free energy density: F 3 NF/r0kBTV,where kB is the Boltzmann constant and T is abso-lute temperature.

8. The Tarazona DFT was selected because it is a simpleand physical approach specifically developed to re-produce the liquid-solid transition in hard spheres.Some DFTs more accurately describe the liquid equa-tion of state but fail to predict the liquid-solid tran-sition; other approaches are more computationallyintensive.

9. N. F. Carnahan, K. E. Starling, J. Chem. Phys. 51, 635(1969).

10. F. Drolet, G. H. Fredrickson, Phys. Rev. Lett. 83, 4317(1999).

11. This approach may also be better suited to examiningmorphologies that lose their symmetry on the scaleof the particle size. See the “crystallization” effect inFig. 3 as an example.

12. Y. Bohbot-Raviv, Z.-G. Wang, Phys. Rev. Lett. 85,3428 (2000).

13. V. Lauter-Pasyuk et al., Physica B 241, 1092 (1997).14. V. Lauter-Pasyuk et al., Physica B 248, 243 (1998).15. B. Hamdoun et al., J. Phys. (France) II 6, 493 (1996).16. J. Huh, V. V. Ginzburg, A. C. Balazs, Macromolecules

33, 8085 (2000).17. The volume fraction has been increased from 0.15 to

0.20 so as to better illustrate “crystallization” effects.Aside from such effects, the morphology is found tobe the same as at 0.15.

18. Supported by the U.S. Department of Energy, NSF,the Army Research Office, and the Engineering andPhysical Sciences Research Council (UK).

12 March 2001; accepted 24 May 2001

Diapiric Flow at SubductionZones: A Recipe for Rapid

TransportPaul S. Hall* and Chris Kincaid

Recent geochemical studies of uranium-thorium series disequilibrium in rocksfrom subduction zones require magmas to be transported through the mantlefrom just above the subducting slab to the surface in as little as ;30,000 years.We present a series of laboratory experiments that investigate the character-istic time scales and flow patterns of the diapiric upwelling model of subductionzone magmatism. Results indicate that the interaction between buoyantlyupwelling diapirs and subduction-induced flow in the mantle creates a networkof low-density, low-viscosity conduits through which buoyant flow is rapid,yielding transport times commensurate with those indicated by uranium-tho-rium studies.

Volcanism associated with the subduction ofoceanic lithosphere is the primary source ofnew continental crust and contributes to thechemical differentiation of the crust-mantlesystem. The mechanism of melt generation atsubduction zones and transport to the surfaceremains uncertain. Proposed models includediapiric flow of a hydrous and/or moltencomponent (1–5), porous flow of magma (6),and melt propagation through fractures (7, 8).Constraints on the melting process at subduc-tion zones come from both geochemical (2, 3,9–13) and geophysical (14, 15) studies. Inparticular, excesses of 238U relative to 230Thobserved in rocks from the Mariana (13) andTonga-Kermadec (11) arcs require the timebetween the fractionation event responsiblefor the disequilibrium (i.e., slab dehydration)and eruption of the magma to be as short as;30 thousand years (ky). Here we report theresults of a series of laboratory experimentsconducted to evaluate the characteristic timescales and morphologies of the diapiric flowmodel.

Conceptual models of mantle diapirism atsubduction zones (1–5) invoke the growth ofgravitational instabilities within a thin buoy-ant layer at the upper surface of the subduct-ing slab to explain arc volcanism (Fig. 1A).Individual instabilities detach from the layerand rise through the mantle wedge as diapirsalong trajectories determined by the balancebetween buoyancy forces and drag from sub-duction-induced motion in the mantle wedge.The diapir model can explain some geophysi-cal observations of magmatism at subductionzones, including the observed regularity ofspacing between discrete volcanic centersalong arcs (d 5 70 km) and the presence andcharacter of secondary, or transverse, volca-

nism at many arcs (1, 16, 17). It also accountsfor many of the petrologic characteristics ofarc rocks (3, 4). Previous laboratory studiesof the diapir model were concerned with thephysics of instability growth from a line ofbuoyant, low-viscosity fluid embedded with-in a stagnant, viscous fluid and focused onspatial variations in the resulting flow (1, 17,18). Our work extends these earlier experi-ments by including the effect of flow in theviscous fluid and focusing on temporal vari-ations in the patterns of buoyant flow.

Results are presented from a series of 10laboratory experiments in which we simulat-ed subduction-driven (matrix) flow with aconcentrated sucrose solution and flow ofbuoyant (hydrated or partially molten) mantlewith a dyed alcohol solution (Fig. 1B). Thebuoyant fluid is introduced from a pointsource, representing what would be a singlesite of instability growth from a linear buoy-ancy source [the distribution of instabilitiesalong a linear buoyancy source was previous-ly investigated and yielded a regular spacingof ;70 km between instabilities (1)]. Thisparameterization of the buoyancy source isnecessary in investigating the temporal evo-lution of the flow because of the experimentaldifficulties inherent in maintaining a constantlinear source of compositionally buoyant flu-id. The flux of buoyant fluid and the velocityof the subducting plate are varied to investi-gate a range in the relative strength of buoy-ant and forced flow. Laboratory and mantlelength scales are related by assuming that thedepth of the buoyancy source in the lab ap-paratus, D (17 cm), corresponds to the uni-form depth of the Wadati-Benioff zone ob-served directly beneath island arcs (125 km).With this fixed length scale, the Mylar veloc-ity can be scaled to a subduction velocity inthe mantle through the dimensionless ratio ofbuoyant flow velocity (from Stokes’ law for aspherical body) to subduction velocity (U0)given by

Graduate School of Oceanography, University ofRhode Island, Narragansett, RI 02882, USA.

*To whom correspondence should be addressed: E-mail: phall@gso.uri.edu

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