Self-Assembly of AmphiphilicNanoparticle-Coil “Tadpole” Macromolecules

Jae Youn Lee and Anna C. Balazs*

Department of Chemical and Petroleum Engineering,University of Pittsburgh, Pittsburgh, Pennsylvania 15261

Russell B. Thompson

Theoretical Division, Los Alamos National Laboratory,Los Alamos, New Mexico 87545

Randall M. Hill

Dow Corning Corporation, Midland, Michigan 48686-0994

Received October 13, 2003Revised Manuscript Received April 2, 2004

Introduction. There has been considerable fascina-tion with the self-assembling behavior of amphiphilicchainlike molecules that range from short-chain sur-factants to high molecular weight block copolymers. Theself-assembly of simple amphiphiles into membranesmay have played an important role in the origin of life.1The self-organization of amphiphiles with more complexarchitectures can lead to a stunning variety of complexmorphologies.2,3 In the case of short-chain surfactants,the equilibrium morphology of the self-assembled sys-tem depends on geometric factors, such as the ratio ofthe “head” to “tail” sizes.4 Here, the headgroups aresmall molecules and the tails are coillike. In the case ofblock copolymers, the structure of the melt depends onthe relative composition of the chains, the degree ofpolymerization, and the incompatibility between thedifferent blocks.

It is intriguing to think of another potentially self-assembling system, one in which the tail is still a coillikemolecule but the attached “head” is a nanoparticle. Inthis scenario, the coil and the surface of the headgroupare incompatible, and this incompatibility could drivethe organization of these “tadpole” molecules intospatially periodic, nanoscale structures. Recent theoreti-cal5 and experimental studies6-9 involving mixtures ofnanoscopic inorganic particles and diblock copolymershave focused on harnessing the microphase separationof the copolymers to template the self-assembly ofnanoparticles and, in this manner, create nanostruc-tured organic/inorganic hybrid materials. The use ofamphiphilic organic/inorganic tadpoles, where the headsare formed from inorganic particles and the tails aresynthetic polymers, could provide an alternative routefor creating such ordered nanocomposites. As we showbelow, the single-tailed tadpoles also display distinctinterfacial activity when blended with diblock copoly-mers. If the heads were covered with multiple, uni-formly distributed hairs, the species would preferen-tially localize in the hair-compatible phase. However,with a single hair, the tadpole behaves like a surfactant,with a head localized in one phase and the tail in theother. These interfacial interactions can be exploited tocontrol the spatial distribution of the tadpoles within acopolymer melt, yielding an additional means of design-ing novel nanocomposites.

One challenge to investigating the properties of thesetadpoles is synthesizing the macromolecules so that the

nanoparticle head is coated with just a single hair.Promising experiments are currently underway to an-chor a single chain onto gold nanoparticles.10 Anotherchallenge is developing a theoretical approach forpredicting the equilibrium structure of the system. Inprevious studies,5 we developed a model that integratesa self-consistent-field theory (SCFT) and a densityfunctional theory (DFT) to investigate the phase behav-ior of mixtures of nanoparticles and diblock copolymers.Experimental studies6 have recently confirmed our“SCF/DFT” predictions on the entropically driven sizesegregation of binary particle mixtures within a diblockmelt.11,12 Experiments7 have also validated our observa-tion that added nanoparticles can promote transitionsbetween the different structures of the diblock copoly-mers.13

In this paper, we extend this SCF/DFT model toexamine the equilibrium structure of a melt of novelamphiphilic “copolymers”, where each copolymer con-sists of a linear flexible chain that is anchored to thesurface of a solid, spherical particle. In this SCF/DFTapproach, the SCFT captures the thermodynamic be-havior of the anchored chain, and the DFT describesthe ordering and steric interactions of the particle.Using this method, we examine the behavior of ABtadpoles, which have an A tail and an incompatible Bhead. We find that the phase behavior of these hybridmolecules differs significantly from that of pure ABdiblocks with the same effective composition. Wealso derive an expression to describe the self-assemblyof ABC tadpoles where the anchored chain is anAB diblock and C is an incompatible particle. Wespecifically focus on the case where C is chemicallyidentical to the B species and find that the ABBmolecules organize in a way that is distinct from theAB tadpoles. Finally, we highlight one example of amixture of AB diblocks and tadpoles to illustrate amethod for tailoring the structure of organic/inorganicnanocomposites.

The Model. A volume V is filled with n tadpolemolecules, where each molecule consists of an A-likelinear, flexible chain that is attached to a B-likespherical, hard particle of radius R. To further charac-terize this system, we define a set of variables thatare specific to the tadpole architecture. We define N/F0to be the volume of the tadpole, where N is thedegree of polymerization of a fictive, linear chainwhose segment volume is F0

-1. The variable f is thevolume fraction of the A-like chain in the molecule. Interms of these definitions, the total volume of eachmolecule is vt ) vA + vR ) N/F0, where the volume ofthe linear chain moiety is vA ) fN/F0 and the volume ofthe sphere moiety is vR ) (1 - f)N/F0 ) 4πR3/3. For agiven volume vt, f and R are not independent, and wespecify R and determine f according to the relation f )1 - (vR/vt). In addition, we describe the incompatibilitybetween A and B species with the Flory-Hugginsparameter ø, which characterizes the interaction be-tween A monomers that are in contact with the Bparticles. Thus, the system is characterized by thefollowing three independent variables: R (or f), N,and ø. Finally, we assume that the system is incom-pressible.

3536 Macromolecules 2004, 37, 3536-3539

10.1021/ma035542q CCC: $27.50 © 2004 American Chemical SocietyPublished on Web 04/20/2004

The following concentration operators describe thedistribution of each species:

where the subscripts A and B denote the A-like coil andB-like sphere, respectively, and FB is the distributionoperator for the centers of the spheres. The rR(s) describethe space curve occupied by the A chains in the Rthtadpole. A unit vector nR is introduced in (3) to indicatethe direction between the end of the coil, which isattached to the surface of the sphere, and the center ofthe sphere of the Rth tadpole. The unit vector for thistadpole is

where ı, j, and k are the three Cartesian unit vectors.The angles θR and φR are measured from the respectivez and x axes in the conventional way for sphericalcoordinates.

Following the procedure in refs 14-16, the partitionfunction for the system of tadpoles subject to the fieldsWi can be written using functions instead of operatorsas

where the functions Φi and FB replace the correspondingconcentration operators and the partition function fora single tadpole Q is given by

We assume that the tails are Gaussian chains so theweighting factor for individual configurations is givenby

in analogy with the properties of the coils in the case ofrod-coil diblocks.15 Here, a is the statistical segmentlength.

From the mean-field approximation of (5), we obtaina free energy expression for the system; we add acorrection term to this expression to account for thesteric interactions between the hard-sphere heads. This

approach follows reasoning described previously17 andgives a free energy of

where Ψ is the Carnahan-Starling free energy perparticle,18 æjB(r) is a “smoothed” sphere density obtainedfrom the Tarazona DFT,19 kB is Boltzmann’s constant,and T is the temperature. The respective fields wi(r) andthe local volume fractions æi(r) are functions for whichthe free energy attains its minimum. We vary the freeenergy with respect to each of the independent functionsto obtain the mean-field equations,14-16,20 which allowus to solve the system self-consistently. As a result, weobtain the densities and fields for various possiblemorphologies, with the lowest free energy phase beingthe equilibrium state.

This model can be easily extended to describe an ABCcopolymer in which an AB diblock is attached to a Ctype sphere. As before, we specify the radius R anddetermine the fraction of the diblock portion f. Anadditional parameter fA, the fraction of the A siteswithin the diblock, is needed to complete the descriptionof the system. The total volume of the molecule is thenvt ) vA + vB + vR, where vA ) fAfN/F0, vB ) (1 - fA)fN/F0, and vR ) 4πR3/3 ) (1 - f)N/F0. The distributionoperators of each species are defined similarly to (1)-(3), and the free energy is written as

where additional terms are introduced to reflect theadditional interactions between the different species.

To obtain solutions to the mean-field equations, weimplement the combinatorial screening technique ofDrolet and Fredrickson.21 In addition, we minimize ourfree energy with respect to the size of the simulationbox.22

Results and Discussion. In this short communica-tion, there is only room to describe illustrative examplesthat capture the unique behavior of these tadpoles; wewill explore the phase space in more detail in futurestudies. Here, we first examine the morphology of a meltof AB tadpoles and then compare this morphology tothe one found for ABB systems, where each moleculeconsists of an AB diblock tail and a B head.

The volume of an AB tadpole is set equal to thevolume of a linear chain whose invariant polymerizationindex Nh ) NF0

2a6 ) 1000, where a is the statisticalsegment length. The sphere radius is set to R ) 0.15R0,where R0 ) aN1/2 is the root-mean-square end-to-enddistance of the Nh ) 1000 chain. Here, we set a ) 1. Wethen calculate f as (F0/N)(N/F0 - 4πR3/3) ) (1 - (4π/3)-(0.15)3Nh 1/2), which yields f ) 0.55. Thus, this systemcorresponds to 55% coil and 45% sphere. The incompat-ibility parameter is set to øN ) 30; at such a large value,we anticipate that system will undergo microphaseseparation.

æA(r) )N

F0∑

R∫0

fds δ(r - rR(s)) (1)

æB(r) ) 1 - fvR

∫|r′|<Rdr′ FB(r′ + r) (2)

FB(r) )N

F0∑R)1

n

δ(r - rR(f) - RnR) (3)

nR ) sin θR cos φRı + sin θR sin φRj + cos θRk (4)

Z ) ∫DΦADFBDWADWBD¥Qn

exp{-F0

N∫dr [øNΦAΦB - WAΦA - WBFB - ¥(1 -

ΦA - ΦB)]} (5)

Q ≡ ∫dnR ∫DrRP[rR;0, f] exp{-∫0

fds WA(rR(s)) -

WB(rR(f ) + RnR)} (6)

P[rR; s1, s2] ) exp{- 32Na2∫s1

s2ds | dds

rR(s)|2} (7)

NFF0kBTV

) -ln(QfNVF0

) + 1V∫dr [øNæA(r) æB(r) -

wA(r) æA(r) - wB(r) FB(r) - ê(1 - æA(r) - æB(r)) +FB(r)Ψ(æj B(r))] (8)

NFF0kBTV

) -ln(QfNVF0

) + 1V∫dr [øABNæA(r) æB(r) +

øACNæA(r) æC(r) + øBCNæB(r) æC(r) - wA(r) æA(r) -wB(r) æB(r) - wC(r) FC(r) - ê(1 - æA(r) - æB(r) -

æC(r)) + FC(r)Ψ(æj C(r))] (9)

Macromolecules, Vol. 37, No. 10, 2004 Communications to the Editor 3537

Figure 1 shows a two-dimensional density contourplot for the coil and sphere portions in the tadpolesystem. The material clearly displays a hexagonal close-packed morphology; the same structure is also observedin the three-dimensional calculations. This finding isrelatively surprising since for a 55/45 A/B composition,a pure diblock melt would exhibit a lamellar morphol-ogy. Thus, it appears that the tadpoles self-organize ina manner that is distinct from these diblock chains. Thisobservation agrees with recent findings from Browniandynamics simulations on nanoparticles functionalizedwith oligomeric tethers.23

In Figure 2, we plot the one-dimensional densityprofiles for the AB tadpoles and the 55/45 AB diblocks.These profiles are not projections of 2D density profilesalong a certain axis. Rather, the density distributionsare obtained by assuming translational invariance alongtwo spatial axes. Although this assumption fails tocapture the correct morphology of the pure tadpolesystem, these 1D plots help in further pinpointingdifferences in the equilibrium structures for the twocases. The plots reveal that there is a greater degree ofoverlap between the A and B fragments in the tadpolesthan in the diblocks. The stretched chains can morereadily segregate from each other than the sphericalhead and tail. Another obvious difference between thetwo systems is that the period of the AB domains ismuch smaller for the tadpoles than the diblocks; this

behavior can be attributed to the steric interactionsbetween the solid spheres. Because of packing con-straints, the solid headgroups are less effective thanchains at associating into large domains; hence, theperiod is smaller in the tadpole system. If B-like chainswhere present in the tadpole melt, these chains couldfill the void space between the hard sphere, and thesystem could, as we will show in the next example, forma lamellar structure. Here, however, there are no Bchains to fill the voids, and the system cannot assembleinto a lamellar morphology but must take on a hexa-gonal structure.

A close look at the distribution of headgroups inFigure 1 seems to indicate that each sphere-rich regioncorresponds to a single sphere of radius R ) 0.15R0.With respect to this unusual feature, it is interestingto contrast the self-assembling behavior of the tadpoleswith the micellization of surfactant molecules with abulky headgroup. In the case of such surfactants insolution, the head-to-tail ratio determines the micellarstructure.4 In the case of a melt of tadpoles with solidheads, steric interactions between the spheres dominatethe behavior of the system, resulting in a possibleformation of “unimolecular micelles”.

To diminish the effect of the steric repulsion betweenthe spheres, we attach an AB diblock chain to a sphere.Here, the B block is anchored to surface of the B-likeparticle. Figure 3 reveals the density profiles for a meltof such molecules, where øN ) 30, R ) 0.1R0, and fA )0.6. The parameters were selected so that the overallAB composition corresponds to one yielding a lamellarmorphology. Here, the A-like coil portion is approxi-mately 52% of the total volume, while the B-like coiland the sphere portions make up the rest. Unlike the

Figure 1. Two-dimensional plots for the tadpole system. Plotsare for øN ) 30, f ) 0.55, and R ) 0.15R0. The plot in (a)represents the distribution of the A tails, and the plot in (b)represents the distribution of the B heads. Light regionsindicate a high density, while dark regions indicate lowdensities. The images show that the system displays a close-packed hexagonal structure.

Figure 2. One-dimensional density profiles for (a) the 55/45tadpole melt and (b) a 55/45 AB diblock melt. For the tadpoles,the parameters are identical to those used in Figure 1. Forthe diblock, Nh ) 1000 and øN ) 30. Solid curves representthe distribution of the A-like species, and the dashed linesmark the distribution of the B-like species.

3538 Communications to the Editor Macromolecules, Vol. 37, No. 10, 2004

AB tadpole case, the ABB tadpoles can aggregate toform a lamellar structure, as shown in Figure 3 andconfirmed by 2D calculations on this system. As notedabove, the B-like chains can fill the void sites betweenthe hard-sphere heads. This behavior is clearly visiblein Figure 3, where the dot-dashed black lines mark thelocation of the B blocks and show that these blocksextend throughout the B particle domains. As a conse-quence, the B domains can form stripes and the entiresystem can exhibit a lamellar phase.

In the final example, we consider a 50/50 mixture ofthe 55/45 AB diblocks and 55/45 tadpoles. The calcula-tion involves adding the SCFT terms for the purediblocks14,17 to the expression in (8). This mixture alsoforms a lamellar structure, and Figure 4 shows the 1Ddensity plots for the system. By comparing this plot withFigure 2a, we can see that the presence of the diblocksleads to sharper, narrower interfaces between the A andB domains. Note that the spheres now essentially forma bilayer within the B domains.

Conclusions. In summary, we have developed anumerical model for describing the phase behavior oftadpole copolymers, which are formed by attaching aflexible chain to a solid, nanoscopic headgroup. Usingthis approach, we have shown that the phase behaviorof tadpole copolymers can differ significantly from thatof compositionally comparable diblocks. This differencein behavior is due to the presence of steric interactionsbetween the hard spheres in the tadpole system. Thesefindings can be exploited to create nanostructured

composite materials that exhibit specific geometries andproperties. For example, a linear polymer can be an-chored to a solid nanoparticle with a specific, desiredindex of refraction, and the resulting spatially periodicsystem can display novel optical properties. If theparticles exhibit unique electromagnetic behavior, theclosely spaced particles in the resultant material canyield optimal electrical or magnetic properties.

As the studies on the ABB system indicated, thearrangement of the nanoparticles can be tailored byanchoring block copolymers onto the surface of thenanoparticles. The addition of AB diblocks to the meltof tadpoles also provides a means of controlling theoverall morphology of the system and the distributionof particles within the material. Further studies arecurrently underway to examine the phase behavior ofmixtures of diblock copolymers and AB or ABC tadpolemolecules. These studies will provide additional guide-lines for creating nanostructured materials with con-trolled morphologies.

Acknowledgment. The authors acknowledge help-ful discussions with Prof. Mark Matsen. Financialsupport from Dow-Corning Corp. is gratefully acknowl-edged.

References and Notes

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Academic Press: London, 1985.(5) Balazs A. C. Curr. Opin. Solid State Mater. Sci. 2003, 7, 27

and references therein.(6) Bockstaller, M. R.; Lapetnikov, Y.; Margel, S.; Thomas, E.

L. J. Am. Chem. Soc. 2003, 125, 5276.(7) Yeh, S.-W.; Wei, K.-H.; Sun, Y.-S.; Jeng, U.-S.; Liang, K. S.

Macromolecules 2003, 36, 7903.(8) (a) Lauter-Pasyuk, V.; et al. Langmuir 2003, 19, 7783. (b)

Lauter-Pasyuk, V.; Lauter, H. J.; Ausserre, D.; Gallot, Y.;Cabuil, V.; Hamdoun, B.; Kornilov, E. I. Physica B 1998,248, 243.

(9) Lopes, W. A.; Jaeger, H. M. Nature (London) 2001, 414, 735and references therein.

(10) Huo, Q. Polymer and Coatings Department, North DakotaState University, private communication.

(11) Thompson, R. B.; Lee, J. Y.; Jasnow, D.; Balazs, A. C. Phys.Rev. E 2002, 66, 031801.

(12) Lee, J. Y.; Thompson, R. B.; Jasnow, D.; Balazs, A. C. Phys.Rev. Lett. 2002, 89, 155503.

(13) Lee, J. Y.; Thompson, R. B.; Jasnow, D.; Balazs, A. C.Macromolecules 2002, 35, 4855.

(14) Matsen, M. W.; Schick, M. Phys. Rev. Lett. 1994, 72, 2660.(15) Matsen, M. W.; Barrett, C. J. Chem. Phys. 1998, 109, 4108.(16) Matsen, M. W. J. Phys.: Condens. Matter 2002, 14, R21.(17) Thompson, R. B.; Ginzburg, V. V.; Matsen, M. W.; Balazs,

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635.(19) Tarazona, P. Mol. Phys. 1984, 52, 81.(20) In solving the mean-field equations, it is necessary to solve

a modified diffusion equation, which is detailed elsewhere.16

The propagators q(r,s) and q†(r,s) which appear in thediffusion equation must have the initial conditions q(r,0) )1 and q†(r,f) ) ∫dn exp{-wB(r + Rn)}. The first conditionis standard, but the second is unique to this architecture.

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MA035542Q

Figure 3. One-dimensional density profiles for ABB tadpolesystems. The parameters are øN ) 30, R ) 0.1R0, and fA )0.6. The A-like coil portion is approximately 52% of the totalvolume, while the B-like coil and the sphere portions makeup 48%. The solid curve represents the distribution of the Ablock, and the dot-dashed curve represents the distributionof the B block in the AB tail; the dashed curve marks thedistribution of the solid B heads.

Figure 4. One-dimensional density profiles for the 50/50mixture of 55/45 tadpoles and diblocks shown in Figure 2. Thesolid curve represents the distribution of the A monomers ofthe diblock, while the dot-dashed curve marks the B mono-mers of the diblock. The distribution of the A tail of the tadpoleis represented by the long-dashed curve, and the distributionof the solid B head is marked by the dotted curve.

Macromolecules, Vol. 37, No. 10, 2004 Communications to the Editor 3539