Origins of Complex Self-Assembly in BlockCopolymers

M. W. Matsen* and F. S. Bates

Department of Chemical Engineering and Materials Science,University of Minnesota, Minneapolis, Minnesota 55455

Received May 21, 1996Revised Manuscript Received August 19, 1996

Amphiphilic molecules are renowned for their abilityto partition chemically immiscible components intonanoscale domains. Often these domains exhibit in-triguing complex periodic geometries with long-rangeorder. Surprisingly, the diverse systems that self-assemble in this manner, surfactants, lipids, soaps, andblock copolymers, exhibit topologically identical geom-etries, suggesting to researchers that a common set ofprinciples govern amphiphilic phase selection. Fromthis association has emerged the belief that constantmean curvature (CMC) interfaces are generally goodmodels for block copolymer microdomain geometries. Bytaking advantage of new developments in polymertheory, we accurately examine this hypothesis for thefirst time, and find it to be wrong. Furthermore, ourstudy reveals new explanations for complex phaseselection that are relevant to numerous block copolymersystems.A block copolymer consists of chemically distinct

polymer chains (i.e., blocks) joined together to form asingle macromolecule. As a consequence of a generaltendency for the blocks to separate, tempered by therestriction imposed by the covalent bonds that connectthem, these molecules exhibit amphiphilic behavior.Even in the simplest case, AB diblock copolymers, a richassortment of ordered phases has been documented.1-8

The composition of the AB diblock (i.e., the volumefraction f of block A) controls the geometry of thestructure (see Figure 1). For nearly symmetric diblocks(f ∼ 1/2), a lamellar (L) phase occurs. For moderateasymmetries, a complex bicontinuous state, known asthe gyroid (G) phase, has been observed in which theminority blocks form domains consisting of two inter-weaving threefold-coordinated lattices.1,2 (Prior to thediscovery of the G phase, a double-diamond (D) struc-ture formed from two fourfold-coordinated lattices3 waserroneously associated with the bicontinuous state inthese materials.4) Another complex structure, theperforated lamellar (PL) phase, occurs when the minor-ity-component layers of the L phase develop a hexagonalarrangement of passages.5 At yet higher asymmetries,the minority component forms hexagonally packedcylinders (C) and then spheres (S) arranged on a body-centered cubic lattice. Eventually, as f f 0 or 1, adisordered phase results.The complete mean field or rather self-consistent

field theory (SCFT) for block copolymers was developedby Helfand and co-workers.9 However, at the time ofits development, it had to be supplemented with ap-proximations limiting its effectiveness. Nevertheless,important advances were made by examining thistheory in the limits of weak10 and strong11 segregation.(The degree to which the A and B blocks segregate isdetermined by the product øN, where ø is the Flory-Huggins A/B interaction parameter and N is the total

degree of polymerization.) The combination of theseworks established that the underlying physics control-ling block copolymer phase behavior involves a competi-tion between interfacial tension and the entropic penaltyfor stretching polymer coils so as to fill space uniformly.The balance determines the equilibrium size of themicrodomains and dictates the geometry of the struc-ture. Although these earlier approaches correctly pre-dicted the classical phases (i.e., L, C, and S),10,11 theyfailed to account for the more recently discoveredcomplex phases (i.e., G and PL).12,13

With new advances,14 it is now possible to implementthe full SCFT. The first calculations14,15 to do so eval-uated the free energies of the structures described aboveand established the phase diagram. This demonstratedthat complex phase behavior occurs in the intermediate-segregation regime as opposed to the weak- and strong-segregation regimes treated by Leibler10 and Semenov,11respectively. For intermediate segregation (e.g., øN )20), the new calculations predict the sequence L f Gf C f S f disordered as f progresses from 1/2 to either0 or 1. Although PL is absent from this sequence, it isnearly stable at the L/G phase boundary, consistent withwhere it is observed experimentally.6 This supportsvery recent experiments indicating that the PL struc-ture is only a long-lived metastable state. The D phaseis clearly unstable, in agreement with current experi-ments.4,6 Given this theoretical accomplishment, wenow probe deeper into the theory14 to examine thephysical factors responsible for complex phase behavior.As described below, the explanation lies in the detailedshape of the dividing interface between the A and Bmicrodomains.Earlier works, such as that of Semenov,11 illustrate

that the phase transitions are driven by a tendency tocurve the interface as the diblocks become asymmetricin composition. The curvature allows the molecules tobalance the degree of stretching between the A and Bblocks. We demonstrate this quantitatively in Figure

* Present address: Polymer Science Centre, University ofReading, Whiteknights, Reading RG6 6AF, UK.

Figure 1. Area-averaged mean curvature ⟨H⟩ as a functionof the A-block volume fraction f for each of the structuresshown schematically calculated using self-consistent mean-field theory.14,15 The stable and metastable states are shownwith solid and dashed lines, respectively, and transitions aredenoted by dots. As the molecules become asymmetric,structures with more curvature are preferred.

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1 by evaluating the area-average ⟨H⟩ of the meancurvature H ≡ 1/2(C1 + C2) for each structure, where C1and C2 are the principal curvatures16 at a given pointon the surface. (Throughout this paper, lengths areexpressed in terms of the statistical end-to-end lengthof an unperturbed diblock.) The trend in Figure 1 isobvious; as f deviates from 1/2, transitions occur tostructures possessing more interfacial curvature. How-ever, based on variations in ⟨H⟩ alone, we might expectthe sequence L f PL f G f D f C f S f disordered,indicating that something in addition to curvature isresponsible for complex phase selection.While the average ⟨H⟩ of the mean curvature controls

the sequence of phases, we will demonstrate that thestandard deviation σH of the mean curvature governsthe phase selection. According to an idea proposed byThomas et al.,8 interfacial tension is a dominating factorin block copolymers, and therefore structures adoptarea-minimizing surfaces of constant mean curvature(CMC),16 which implies σH ≈ 0. Here, we demonstratethat there is a second equally important factor, packingfrustration, which has been identified earlier by Grunerand co-workers for lipid-membrane forming systems17,18and subsequently suggested as a mechanism in blockcopolymers.2,7 For block copolymers, this mechanismtranslates into a tendency to form domains of uniformthickness so that none of the molecules are excessivelystretched, which causes σH to deviate from zero. Al-though this effect has been observed in ABC triblockcopolymers,19 it is typically regarded to be of minorimportance. Below we show that this is not true, andthat in fact it plays an instrumental role in complexphase selection.The above mechanisms are most easily discussed in

terms of the C phase. Figure 2a shows the A-segmentprofile φA through a cylinder for two degrees of segrega-tion; this demonstrates both the increase in domain sizeand the decrease in interfacial width as øN increases.The interfacial surface, given by φA ) 1/2 (see Figure2a), is in general not circular.19 It is well approximatedby r(θ) ) r0(1 + δ cos(6θ)), where δ measures thedeviation from CMC. While the interfacial tension andpacking considerations in the minority domain favor δ) 0, the majority domain prefers δ > 0 so as to producea more uniform thickness (i.e., L1 ≈ L2 in Figure 2b).However, the frustration in the majority domain is smalland δ only deviates slightly from zero (see Figure 2c).In general, all the classical phases, L, C, and S, allowthe molecules to pack efficiently and therefore do notexhibit significant variations from CMC. Contrary topublished speculations,8 the surface becomes less CMC-like at strong segregations as demonstrated by Figure2c. The explanation is simple; as øN increases, themolecules become highly stretched relative to theirunperturbed end-to-end statistical length, making itmore difficult to fill the corners of the Wigner-Seitz cell.This observation that increasing øN amplifies the pack-ing frustration will explain the absence of complexphases in the strong-segregation regime.Unlike the classical phases, the complex phases suffer

from high degrees of packing frustration, producinglarge deviations from CMC. Figure 3 shows the inter-facial curvature over elementary units of the C, G, PL,and D structures calculated at the C/G transition inFigure 1. From the distribution of H, shown with thecolor scale, the average ⟨H⟩ and standard deviation σHare calculated for each structure. While the C structureis nearly CMC (i.e., σH ≈ 0), the complex phases all

exhibit significant variations in H. The quantity σHdirectly reflects the inability of a structure to simulta-neously minimize surface area and packing frustration,and therefore is correlated to its stability. We attributea large portion of σH to frustration in the minorityblocks, which have to fill the space at the center of theconnectors in Figure 3. Naturally, this frustration islargest for the four-connector (i.e., the D phase). Webelieve G is favored over PL because its majority domainis less frustrated. Support for these deductions comesfrom calculations on diblock/homopolymer blends.20Adding homopolymer to the minority domain relievespacking frustration and can cause a transition from Gto D. Similarly, adding homopolymer to the majoritydomain of G can stabilize PL. In Figure 4, we demon-strate that this increase in stability is accompanied bya decrease in σH; further explanations for the trends inFigure 4 will be provided in a future publication. Wenote that this reduction in packing frustration closelyresembles what is encountered with membrane-forminglipids and surfactant solutions.17 Finally, the theoreti-cal13,15,21 and experimental7 evidence that complexphases are unstable at strong segregations is rational-ized by recalling that increasing øN exacerbates thepacking frustration.

Figure 2. (a) A-segment profile across a minority domain ofthe cylinder (C) phase for f ) 0.3378 at two degrees ofsegregation, øN. The location of the interface defined by φA )1/2 is indicated with dots. (b) TheWigner-Seitz hexagonal unitcell showing the interface between the A and B domains, anda representative diblock copolymer molecule. The interfacialshape is, in general, well described by r(θ) ) r0(1 + δ cos(6θ)).The stretching energy of the B domain prefers δ ≈ 0.03 asillustrated in the figure so as to produce a relatively uniformthickness (i.e., L1 ≈ L2). On the other hand, interfacial energyfavors δ ) 0 (i.e., a CMC surface). (c) The values of δ thatminimize interfacial tension and packing frustration for vari-ous segregations at f ) 0.3378. The small values reflect a nearabsence of packing frustration in the C phase.

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0.8

H 0.6

0.4

0.0

C(H)= 0.970on = 0'003

G(H;= 6.700oH = 0.121

(H)= 0.615ou = 0.107

(H)= 0.555oli = 0'113

(H;= s.6taoH = 0.303

PL(H;= 6.6ruoH = 0.146

Communications to the Editor 7649

(H)= 0.4i9ou= 0'126

(FI)= 0.397on = 0'190

(H;= g.otnoH = 0.105

D(H)= 0.141oH = 0.311

Figure 3. Interfacial surfaces associated with elementarvunits of the C, G, PL, and D structures calculated at the C/Gp_hase boundary (.yN:20 andf :0.3378). For each structure,the distribution of mean curvature H over the surface isindicated using the color scale, and the area-average (Ii) andstandard deviation ou of H is provided. The large values ofoH reflect large degrees ofpacking frustration peiturbing theinterface away frorn CMC. Note that the three-connector ofthe G structure is planar, but the connector ofthe pL phaseis slightly lorrplanar, which produces an asymmetry beiweenthe top and bottom sides. The bar at the 6ottom reDresentsone unit of length.

While the individual mechanisms that control com-plex phase selection are conceptually familiar,8,17,18 16the best of our knowledge, there are no prior first-principles calculati ons thorou ghly examining their rel a-tive importance. As a result, packing frustration hasfailed to receive proper recognition for its influence onblock copolymer phase behavior. The full SCFT permitsthis study because it allows each microstructure toadjust its interface so as to minimize the energyassociated with the combination of interfacial area andpacking frustration. Other calculations13,21,22 typicallyrequire a priori assumptions regarding interfacial shape,while those that do allow the segment profiles toadjuslro'rz are unable to treat experimentalfu relevantdegrees ofsegregation where the interface is suffrcientlydeveloped to apply these concepts. Based on our SCFfcalculation, packing frustration prevents the stabilityofthe D and PL phases, and in general the stability ofall complex phases in the strong-segr:egation limit. Wehave also illustrated which domains of the D and pLphases are most frustrated and how this can be relievedwith the addition of an appropriate homopolymer. Suchideas will guide future efforts to stabilize new complex

t'igure 4. Plots similar to Figure B using the same color scale,but caiculated with added homopolymer. (The degree ofpolymerization of the homopolS,'rner-equals that of the d]block.)Surface units on the left each containabotft20Vo B homonolv-mer while those on the right have about 10% A homopolymer,which segregates to the majority and minority do*"i"s,respectively. In each case, structures are compaied in the i* G two-phase coexistence region at yN : 20.-

phases-in block copolymer blends, and may shed ad-ditional light on the role of molecular frustration inordered soft materials in general.

Acknowledgment. We thank G. H. Fredrickson, S.M. Gruner, D. A. Hajduk, and M. A. Hillmyer for criticalreadings of the manuscript. This work was supportedFy- Lir" Minnesota Supercomputer Institute and by theNSF (Grant DMR 94-05101).

References and Notes(1) Schu?, M. F.; Bates, F. S.;Almdal, K.; Mortensen ,K. phys.

Reu. Lett. 1994. 73. 86.i2) Hajd_uk, D^. Al Harper, p. E.; Gruner, S. M.; Honeker, C.

C.r Kim, G.; Thomas, E. L.; Fetters, L. J. Macromolecules1994.27. 4063.

(3) Thom_as, E L.;Alward, D. B.;Kinning, D. J.; Martin, D. C.;. Handlin, D. L.; Fetters, L. J. MacromoTecules 19g6, Ig,2Ig7'.{4) Hajduk, D. Al Halper, p. E.; Gruner, S. M.; Honeker, C.

C.; Thomas. E. L.; Fetters, L. J. Macromolecules lgg1,'2g,2570.

(5) Hamley,,I. W.; Koppi, K. A.; Rosedale, J. H.; Bates, F. S.;.^ Almdal, K.; Mortensen, K. Macromolecutes 1rlg},26',5gig'.(6) Bates, F. S.; Schulz, M. F.; Khandpur, A. K.; Frirster, S.;

R99eda1e, J. H.; Atmdal, K. ; Mortensen, K. Faraday Disiuss'.L994. 98.7.

(7) Hajduk, D. A.; Gruner, S. M.; Rangarajan, P.; Register, R.A.; Fetters, L. J.; Honekar, C.; Albalak, R. J.; Thomas, E.L. Macromolecules 1994, 27, 490.

(8) Thomas, E. L.; Anderson, D. M.; Henkee, C. S.; Hoffman,D. Nature 1988, 334, 598.

(9) Helfand, E. J. Chem. Phys. 1975, 62, 999.(10) Leibler, L. Macromolecules 1980, 13, 1602.(11) Semenov, A. N. Sov. Phys. JETP 1985, 61, 733.(12) Hamley, I. W.; Bates, F. S. J. Chem. Phys. 1994, 100, 6813.(13) Fredrickson, G. H. Macromolecules 1991, 24, 3456. Olm-

sted, P. D.; Milner, S. T. Phys. Rev. Lett. 1995, 74, 829.(14) Matsen, M. W.; Schick, M. Phys. Rev. Lett. 1994, 72, 2660.(15) Matsen, M. W.; Bates, F. S.Macromolecules 1996, 29, 1091.(16) Anderson, D. M.; Davis, H. T.; Scriven, L. E.; Nitsche, J. C.

C. Adv. Chem. Phys. 1990, 77, 337.

(17) Gruner, S. M. J. Phys. Chem. 1989, 93, 7562.(18) Anderson, D. M.; Gruner, S. M.; Leibler, S. Proc.Natl. Acad.

Sci. U.S.A. 1988, 85, 5364. Turner, D. C.; Gruner, S. M.;Huang, J. S. Biochemistry 1992, 31, 1356.

(19) Gido, S. P.; Schwark, D. W.; Thomas, E. L.; Concalves, M.Macromolecules 1993, 26, 2636.

(20) Matsen, M. W. Phys. Rev. Lett. 1995, 74, 4225; Macromol-ecules 1995, 28, 5765.

(21) Likhtman, A. E.; Semenov, A. N.Macromolecules 1994, 27,3103.

(22) Anderson, D. M.; Thomas, E. L. Macromolecules 1988, 21,3221. Olmsted, P. D.; Milner, S. T. Phys. Rev. Lett. 1994,72, 936.

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