Frank S. Bates- Block Copolymer Thermodynamics: Theory and Experiment

8/3/2019 Frank S. Bates- Block Copolymer Thermodynamics: Theory and Experiment

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Annu. Rev. Phys. Chem.1990. 41:525-57Copyright1990by AnnualReviews nc. All rights reserved

BLOCK COPOLYMERTHERMODYNAMICS:Theory and ExperimentFrank S. BatesDepartment of Chemical Engineering and Materials Science,University of Minnesota, Minneapolis, Minnesota 55455Glenn H. FredricksonAT&T ell Laboratories, Murray Hill, NewJersey 07974KEYWORDS:rder and disorder.

INTRODUCTIONBlock copolymers are macromolecules composedof sequences, or blocks,of chemically distinct repeat units. The development f this field originatedwith the discovery of termination-free anionic polymerization, which madepossible the sequential addition of monomers o various carbanion-ter-minated ("living") linear polymer chains. Polymerization of just two dis-tinct monomerypes (e.g. styrene and isoprene) leads to a class of materialsreferred to as ABblock copolymers. Within this class, a variety of molec-ular architectures is possible. For example, the simplest combination,obtained by the two-step anionic polymerization of A and B monomers,is an (A-B) dioblock copolymer. A three-step reaction provides for thepreparation of(ABA) r (BAB) riblock copolymer. Alternatively, "living"diblock copolymers can be reacted with an n-functional coupling agent toproduce (AB)n star-block copolymers, where n = 2 constitutes a triblockcopolymer. Several representative (A-B)n block copolymer architectures

~ Present address: Department of Chemical and Nuclear Engineering, University of Cali-fornia, Santa Barbara, California 93106.525

0066-426X/90/11014)525502.00

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526 BATES & FREDRICKSONare illustrated in Figure 1. As a consequence f the "living" nature of thesereactions, the resulting block and overall molecular weight distributionsare nearly ideal, i.e. Mw/Mr~ 1.1, where Mw nd MN epresent the weightand number-average molecular weights, respectively.Since the original studies of anionic block copolymerization n the 1950s(1, 2) a variety of new polymerization mcthods e.g. condensation, Ziegler-Natta, etc.) have contributed to an expanding number of block copolymerclasses (e.g. ABC) nd novel architectures (e.g. graft-block). Althoughsome of the developments have resulted in important new materials (e.g.polyurethanes), anionic polymerization remains the only viable methodfor

(A-B)n Block Copolymer Architectures

k.. J,~ triblock

Figure 1 Schematic illustration of several (A-B), type block copolymer architectures. Solidand dashed lines represent A and B block chains. The n = 1 and n = 2 architectures arecommonlyeferred to as diblock and triblock copolymers, while n __ 3 are denoted starblockcopolymers.


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BLOCK COPOLYMER THERMODYNAMICS 527producing monodisperse block copolymers with well-defined architectures.Becausecurrent theories deal almost exclusively with model(A-B), type ma-terials, we have restricted our attention in this review o studies based solelyon this class of anionically polymerized block copolymers see Figure 1).The phase behavior" of undiluted (bulk) (A-B), block copolymersdetermined by three experimentally controllable factors: the overall degreeof polymerization N, architectural constraints characterized by n and thecomposition f (overall volume fraction of the A component), and theA-B segment-segment (Flory-Huggins) interaction parameter ~. [In thepresent review we use the terms monomerand segment interchangeablyto implystatistical segment 3).] The irst two factors are regulated throughthe polymerization stoichiometry and influence the translational and con-figurational entropy, whereas the magnitudeof (the largely enthalpic) ~determined by the selection of the A B monomer air. Wenote that forall the materials considered in this review, the interaction parameter has thetemperature dependenceZ ~ aT- +/Y, where ~ > 0 and/~ are constants forgiven values of f and n. Since the n = 1 case has received the mostcomprehensive heoretical treatments and because the above factors quali-tatively influence phase behavior independent of n, our introductoryremarks and muchof this text focus on diblock copolymers.At equilibrium, a dense collection of monodisperse diblock copolymerchains will be arranged in minimumree energy configurations. Increasingthe energy parameter Z (i.e. lowering the temperature) favors a reductionin A B monomer contacts. If N is sufficiently large, this may beaccomplished with some oss of translational and configurational entropyby local compositional ordering as illustrated in Figure 2 for the symmetric

ORDERED DISORDERED

," k~] ,, ~2 JUNCTION % ., ~.,

t .... % ,, , rFigure 2 Schematic representation of order and disorder in a symmetric (f = 1/2) diblockcopolymer showing lamellar order.


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528 BATES & FREDRICKSONcase f--0.5. (Here we note that block crystallization, which can alsooccur, lies outside the scope of this paper; we consider only amorphouscopolymers.) Such local segregation is often referred to as microphaseseparation; macroscopicphase separation is impossible in a single-com-ponent block copolymer. Alternatively, if either 2 or N is decreasedenough, the entropic factors will dominate, leading to a compositionallydisordered phase. Since the entropic and enthalpic contributions to thefree energy density scale as .N- ~ and ~, respectively, it is the product zNthat dictates the block copolymer phase state. For f= 0.5, the transitionbetween the ordered and disordered states occurs when zN/n ~ 10, asdiscussed below.Two imiting regimes have been postulated to exist in the diblock copoly-mer phase diagram, as illustrated in Figure 3. For ZN


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BLOCKOPOLYMERHERMODYNAMICS 529turbed, the microdomain period scales as N1/2, and the ordered com-position profile is approximately inusoidal. We hall refer to such a regimeas the weak seyre#ation limit (WSL). Current order-disorder transition(ODT) theories are based on this WSL ssumption because it greatlysimplifies calculations, although strict adherence of experimental systemsto the WSL ostulates is still not established. The second limiting regimeof phase behavior is referred to as the strong segregation limit (SSL) andcorresponds to the situation of ~N>> 10. In this regime, narrow interfacesof width (5) a~-~/2 separate well-developed, nearly pure A and B micro-domains. The interaction energy associated with A-Bcontacts is localizedin these interfacial regions; the system would like to minimize the totalarea of such interface, but must do so under the constraint of incom-pressibility and with the entropic penalty of extended chain configurations(5, 6). These opposing forces lead to perturbed chain configurations andmicrodomain imensions (periods) that scale as D ,,~ aN2/39~/6.Since most theories and experiments dealing with block copolymer phasebehavior can be categorized as either WSL r SSL, we have organized thepresent review under these general headings. As is discussed further below,this classification schememaybreak down n the ODTegion. Regardless,it proves most convenient o treat the transition region in the WSLection.The preparation of model (undiluted) block copolymers for the purposeof studying the order-disorder transition (ODT) also referred to as themicrophase separation transition (MST)] is complicated by the limitedrange in zN afforded by experimentally accessible temperatures. In orderto overcome this difficulty, researchers often add modest amounts of aneutral solvent to the bulk material, thereby diluting the A-Bcontacts.Here a neutral solvent is defined as one that showsno preference for eitherblock type. In general, such concentrated solutions behave much ike thebulk materials, with ~ replaced by an effective interaction parameter thatis proportional to the copolymer concentration. Thus, we have includedthis restricted group of block copolymer solution studies in the presentreview; semidilute and dilute copolymer olutions fall outside the scope ofour review.This review is organized into four sections, each dealing with theory andexperiment. In the first and second sections we cover recent developmentsin the strong and weak egregation limits, respectively. Limitations in sizeconstrain us to significant developments hat have occurred within roughlythe past decade. Prior advances in block copolymers are documented nearlier reviews (7-12). We urther restrict ourselves to issues relatedblock copolymer thermodynamics, leaving a vast literature on copolymerdynamics to a future reviewer. In some instances, however, dynamicalproperties and measurements are inextricably coupled with thermo-


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530 BATES & FREDRICKSONdynamic properties. In these circumstances we have not attempted toseparate the issues. Our final topic, addressed in the third section, involvesblock copolymer surfaces. Recent experimental and theoretical advancesin this emerging area have added a new dimension to the field of blockcopolymer thermodynamics. A discussion and outlook section is devotedto assessing recent progress in this field and to speculating on futuredirections.STRONG SEGREGATION LIMIT (SSL)ExperimentUntil roughly a decade ago transmission electron microscopy (TEM)wasthe preeminent experimental technique for studying block copolymerstructure. The combination of relatively large monodisperse micro-structures and efficient heavy-metalstaining techniques (e.g. osmiumetra-oxide) produced truly spectacular electron micrographs of ordered phasesin polystyrene-polydiene block copolymers. Five ordered phases wereidentified in the strong segregation limit. Two ypes of spherical andcylindrical microstructures, as well as a lamellar morphology see Figure4), were shown to exist within well-defined composition ranges in closeagreement with Helfands theoretical predictions (5). During the pastdecade, our analytical capabilities have been greatly enhanced by thedevelopment of small-angle scattering techniques that complementadvances in TEM nd provide access to new thermodynamic and fluc-tuation quantities. The topics covered in this section reflect these recentdevelopments.Hashimotoand co-workers have played a leading role in the application

SSL Morphologies

PS PS PS PS, PI PI PI PISpheres Cylinders GBDD Lamellae OBDD Cylinders SpheresI I I I I I~s 0.17 0.28 0.34 0.62 0.66 0,77forPS-PT diblock copolymers

Figure 4 Strong segregation limit (SSL) equilibrium morphologies for (A-B). type blockcopolymers. The order-order transition compositions shown apply to polystyrene-poly-isoprene diblock copolymers where ~bs corresponds to the polystyrene volume fraction.


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BLOCK COPOLYMER THERMODYNAMICS 531of small-angle X-ray scattering (SAXS) o the investigation of blockcopolymer thermodynamics. In a seminal series of publications (13-18)these authors used SAXS nd TEM o explore microdomain size andpacking, and interfacial mixing in a modelset of polystyrene-polyisoprenediblock copolymers. The quantitative measurement f the interfacial thick-ness between microphases represented an important development, par-ticularly in the light of theoretical predictions by Helfand & Wasserman(5) regarding this parameter. Following procedures developed by Ruland(19), Vonk(20), and others (21), Hashimoto et al evaluated thescattering wavevector region of their SAXS ata (the Porod regime) anddetermined an interfacial thickness t = 20_ 5 ,~ independent of micro-structure (lamellar or spherical) and molecular weight. A subsequentSANS tudy of polystyrene-polybutadiene diblock copolymers by Bateset al (22) producedessentially the same conclusions regarding interracialthickness. [These authors also determined body-centered-cubic packing ofspherical polybutadiene domains (23) in agreement with the weak seg-regation prediction of Leibler (see WSLheory section). However, hispacking symmetrymay not be universal, since Richards & Thomason 24)have deduced a face-centered-cubic symmetry for polystyrene-poly-isoprene diblock copolymers containing polystyrene spherical micro-domains, also based on SANSmeasurements.] In independent SAXSmeasurementson polystyrene-polybutadiene diblock and triblock copoly-mers, Roeet al (25) found 10 ~< t Toox (which we define as being in theWSL) uggesting the presence of microdomain structure within the dis-ordered state.] All these small-angle scattering results for polystyrene-polydiene block copolymers are in reasonably good agreement withthe prediction of Helfand & Wasserman 5), t ~ x/~az- 1/2 ~ 23/~ (seeSSL-Theoryection), where a is the statistical segment ength.Although small-angle scattering experiments are capable of establishinga precise characteristic interfacial thickness, they are not able to dis-criminate between different mathematical expressions for the interfacialprofiles due to limitations set by backgroundand domain cattering. Spon-taket al (26) have attempted to address this deficiency by measuring theinterfacial composition profile directly via TEM.This method requiresuse of extremely thin osmium-stained sections, which raises questionsregarding stain-induced local segregation and swelling. Nevertheless,Spontak et al report interfacial composition profiles with characteristicthicknesses, t ,,~ 26/~, slightly greater than were found by the small-anglescattering Porod analysis.Oneof the most exciting discoveries within the strong segregation regimein recent years is the ordered bicontinuous double diamond (OBDD)


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532 BATES & FREDRICKSONmorphology, depicted in Figure 4 along with the five originally recognizedequilibrium morphologies. A representative electron micrograph of thismorphology is shown in Figure 5 . To our knowledge, the earliest publishedTEM picture of the OBDD phase was obtained from a polystyrene-poly-diene star-block copolymer and reported by Aggarwal(27) as the wagonwheel morphology. Subsequently, Thomas and co-workers (28, 29)initiated a research program aimed at cllucidating the solid state mor-phology of well-defined star-block copolymers prepared by L. J. Fetters(30). For a narrow range of experimental conditions including armnumber, molecular weight, and composition (z 6% by volume poly-styrene), Thomas et a1 found a novel bicontinuous morphology that didnot conform to the well-known ordered phases. Soon thereafter cametheir dramatic identification of the now well-known OBDD phase (31),consisting of two continuous interpenetrating diamond (tetragonally co-ordinated) networks of polystyrene rods embedded in a continuous poly-isoprene matrix. Evidence for this structure was provided by the combineduse of SAXS and TEM. Small-angle scattering reflections from the orderedlattice were instrumental in identifying the exact lattice type and spacegroup, which was then correlated with numerous tilted TEM images basedon computer generated two-dimensional crystallographic projections.

Following these publications on star-block copolymers, Hasegawa et a1(32) reported an equivalent ordered phase, denoted the tetrapod-networkstructure in polystyrene-polyisoprene diblock copolymers, thus demon-

A BFigure 5 (u) Representative transmission electron micrograph of the ordered bicontinuousdouble diamond (OBDD) morphology. This image was obtained from an osmium stainedpolystyrene-polyisoprene block copolymer. (b) A computer simulated projection of theOBDD structure. (Provided by E. I,.Thomas.)


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BLOCK COPOLYMER THERMODYNAMICS 533strating the occurrence of the OBDDhase in simple linear block archi-tectures. [Because these diblock copolymers contained polystyrene as themajor component (62-66% by volume), the two interpenetrating networkswere composedof polyisoprene, i.e. the "tetrapod-network structure" isactually an inverted version of the original OBDDhase identified byThomas t al.] Both research groups evaluated the stability of this mor-phology by preferential solvent casting and annealing experiments. For anarrow range of compositions, which varies slightly with chain archi-tecture, the OBDDhase appears to be the equilibrium state; for poly-styrene-polyisoprene diblock copolymers this occurs for polystyrenevolumefractions of 0.28-0.34 (polystyrene double diamond tructure) and0.62-0.66 (polyisoprene double diamondstructure) as indicated in Figure4 (32, 33).Most recently, Thomasand co-workers have explored the underlyingdriving forces for the formation of the OBDDmicrostructure, demon-strating that it belongs to a class of geometrical structures characterizedby constant mean curvature (CMC) urfaces (34). Several TEMmagesnew block copolymer morphologies were also presented and shown tobelong to a family of periodic area-minimizingsurfaces that have attractedthe attention of mathematicians for over a century. These elegant studieshave revealed a beautiful physical manifestation of abstract mathematicalconcepts, madepossible through the careful control of the three molecularparameters described in the introduction.Central to the theory of strong segregation in block copolymers isthe concept of an extended block conformation, made necessary by thecombined onstraints of block joint localization at a narrow interface andan overall uniform density (see following section). The effects of extendedchain conformation are most readily observed in the molecular weightdependence of the periodic lattice spacing and domain dimensions,D ~ N~, where 6 ~ 2/3 in the SSL versus 6 = 1/2 for unperturbed chainstatistics (assumed) in the WSL.With the advent of small-angle neutronscattering (SANS), he direct experimental determination of block chainstatistics is now ossible.

Richards & Thomason (35) reported the first SANSmeasurementson mixtures of partially labeled and unlabeled polystyrene-polyisoprenediblock copolymers. The SANS attern contained two scattering con-tributions, one deriving from single-block scattering from within the poly-styrene spherical domains (containing 4%deuterated polystyrene blocks)and a second associated with domain scattering. Richards & Thomasonestimated the domain scattering contribution based on the SANS atternobtained from an unlabeled sample, and subtracted it from the totalscattering intensity to arrive at an estimate for the polystyrene block


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534 BATES& FREDRICKSONdimensions within the spherical domains. Unfortunately, this subtractionmethod is extremely sensitive to small differences in molecular weightand composition between the labeled and unlabeled polymers, and thissensitivity limits the practical application of the method.Hadziioannouet al (36) published the first SANS tudy of block chaindimensions in a tamellar microstructure. By mixing small amounts ofpartially deuterated polystyrene-polyisoprene block copolymer (per-deuterated polystyrene block) with equal molecular weight unlabeledmaterial, these researchers created scattering contrast within the poly-styrene domains that could be used.to determine the average block con-formation within the lamcllae. As with the previous case, however, for theconcentrations of labeled polymer used by Hadziioannou et al there is astrong scattering component due to interdomain interference that domi-nates the scattering intensity when the neutron beam s directed parallelto the plane of the lamellae. For unorientcd (i.e. "polycrystalline") samples,this interference effect nearly completely obscures single block scattering.Hadziioannou t al partially rectified this problemby shear-orienting theirmaterial, thereby providing a purely perpendicular incident neutron beamgeometry. This eliminates the interlamellar scattering component, andallows a direct determination of chain dimensionsparallel to the lamellae.Perpendicular dimensions, however, could not be measured. Their studyindicated a significant lateral contraction of the block chain dimensionsparallel to the lamellae, relative to the unconstrainedsize.The topic of single chain scattering in undiluted block copolymers wastreated theoretically by Jahshan & Summerfield(37) and Koberstein (38)in the early 1980s. These authors recognized that by mixing specifiedamountsof partially labeled and unlabeled chains, the contrast factor (i.e.the scattering length density for neutrons) could be matched betweenmicrophases, thus eliminating domain scattering. Within a microdomain,however, isotopic labeling wouldgive rise to single block scattering. Thiscontrast matching technique was first demonstrated by Bates et al (39)a polystyrene-polybutadiene diblock copolymer system in which sphericalmicrodomains were prepared with 16% perdeuterated and 84% hydro-genous polybutadiene blocks. The method proved to be quite effective ateliminating domainscattering, and for low molecular weight polybutadieneblocks revealed an overall radius of gyration in agreement with unper-turbed dimensions. However, as the molecular weight was increased, theapparent block dimensions became unreasonably large; this behavior cannow be attributed to small deviations from the exact contrast matchingcondition or to fluctuation effects (see below).The most recent studies of block chain statistics have madeuse of thecontrast matching technique with the lamellar morphology. This corn-


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BLOCK COPOLYMER THERMODYNAMICS 535bination affords access to both the lateral (parallel) and perpendicularcomponents of the chain conformation within the microdomain space.SANStudies by Hasegawa t al (40, 41) and Matsushita et al (42) demon-strate that even at the theoretical contrast matching condition, someresidual domainscattering remains due to slight composition fluctuationswithin the isotopically mixed microphase; increasing molecular weightexacerbated this problem, as inadvertently found by Bates et al (39).Nevertheless, contrast matching reduces the interference effects by up totwo orders of magnitude, thereby facilitating the determination of theperpendicular block dimension. Hasegawa t al (40, 41) report polystyreneblock chain dimensions parallel and perpendicular to the lamellae to beapproximately 70% and 160%of the unperturbed dimensions, respec-tively, for nearly symmetric polystyrene-polyisoprene diblock copolymerswhen Mw= (7.5-- 10) x 104 g/mole.TheoryBy the middle of the 1970s, the physical principles that govern the micro-domain period and the selection of ordered phases in the SSL had beenwell-established by pioneering studies of Meier (43), Leary & Williams(44), and Helfand & Wasserman (5, 45a,b). Most notably, HelfandWassermandeveloped a self-consistent field theory that permits quan-titative calculations of free energies, compositionprofiles, and chain con-formations. They identified the three principal contributions to the freeenergy in the regime zN >> 10 as arising from (a) contact enthalpy in thenarrow interfaces between nearly pure A and B microdomains, (b) entropyloss associated with extended chain configurations to ensure incom-pressibility (i.e. stretching free energy), and (c) confinement ntropy duelocalization of the block copolymer oints (covalent bonding sites betweenblocks) to the interfacial regions. Helfand & Wassermanhowed hat thesenarrow interfaces have characteristic thickness az-1/2, and by numericalsolutions of the self-consistent field equations proposed that the micro-domain period scales (asymptotically for N~ ~) as D ~ aN6g~, with6 ~ 9/14 ~ 0.643 and ~t ~ 1/7 ~ 0.143. In this asymptotic limit the con-finement entropy of the j unction is negligible comparedwith the stretchingentropy. Helfand & Wasserman lso developed numerical techniques forcalculating the phase diagram in the $SL, and located the (virtually tem-perature independent) compositions that delimit the thermodynamic tab-ility of spheres, cylinders, and lamellae. These compositions are in goodagreement with experimental determinations of the phase boundaries,although we note that the OBDDhase (having not yet been discovered)was not included in the free energy competition.Although the Helfand-Wassermanheory is believed to contain all the


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536 BATES& FREDRICKSONproper physical ingredients necessary to describe the SSL, its practicalapplication has been hindered because of the requirement of rather difficultnumerical analysis. The theoretical advances of the past decade have beenfocused on developing analytical methods for estimating the free energyin the asymptotic limit 2N-~, ~. Probably the most influential of thesemodern studies was a paper by Semenov 6), which addressed a diblockmelt in the SSL. Semenov rgued that because the copolymers are stronglystretched in this regime (see previous section), the required configurationalintegrals are dominatedby the classical extremum f the energy functional(Hamiltonian). The extremum path corresponds to the most probableconfiguration of a copolymerblock as it extends from an interface into amicrodomainand experiences the chemical potential field produced by thesurrounding molecules. Moreover, Semenov howed that the differentialequation describing this path (which resembles the equation of motion ofa classical particle) is analytically soluble, even when he chemicalpotentialis determined self-consistently to maintain constant density of copolymersegments. This solution indicates that the copolymers are stretched non-uniformly (along their contours) as they enter into the microdomainsandpredicts that the chain ends are distributed at excess in the domain nteriors.The classical mechanical analogy identified by Semenov as been furtherclarified by Milner, Witten & Cates (46, 47a,b) and was extensivelyexploited by these authors to treat related problems of grafted polymerbrushes and surfactant interfaces. To appreciate the reduction in com-plexity afforded by this method, we note that the Helfand-Wassermanapproach corresponds to the solution of a time-dependent problem inquantum mechanics, whereas the Semenov-Milner-Witten-Cates approachrequires only the solution to the classical limit of this problem. Of course,the latter approach is only legitimate under conditions of strong chainstretching.It is somewhat urprising that in spite of the significant chain defor-mations predicted for the SSL, the domain(stretching) contribution to thefree energy per chain was found by Semenov o have the same scaling asfor a Gaussian chain, namely Fdomain/kT~ D2/a2N,where D is the domainperiod. It is only the constant prefactor omitted from this expressionthat reflects the nonuniformstretching and distribution of ends. In theasymptotic limit 2N--r ~, the domain free energy is balanced by theinterfacial energy, which (per chain) is given by (5, 6, 45a,b) Finterface/kT~ 7a ~ Na~.~/2/D. Here we have inserted well-known results for theinterfacial tension, 7 ~ ~.~/2a-2, and for the area per chain, a ,-, Na3/D.Bybalancing the two free energy contributions, Semenovs rediction for thedomainperiod is recovered, D ~., aN2/3Z~/6.This result is believed to beasymptotically correct for large incompatibility, as the fluctuations about


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BLOCK COPOLYMER THERMODYNAMICS 537the classical path are O(N/2) and thus can be neglected for N-, oe. Theslight differences of these exponents from those of Helfand & Wassermancan be traced to the Helfand-Wassermanumerical estimate of Fdomai,. Inparticular, these authors foundFdom,in/kT~ (D/aN1/2)2"5, based on numeri-cal calculations that extended only to D/aN/2 ~ 3. We uspect that if thenumerics had been carried out to D/aN~/~ >> 1, the asymptotic scaling ofSemenov would have been obtained. Finally, we note that Semenovstheory also provides estimates of the compositions that delimit the variousordered phases. As with the Helfand-Wasserman theory, these com-positions are predicted to be tcmpcrature-independent.Another notable SSL theory of the past decade is that due to Ohta &Kawasaki (48a,b). Like the Helfand-Wasserman and Semenov theories,the approach of Ohta & Kawasaki was field-theoretic, although Ohta &Kawasakiemployedonly a single scalar field describing the compositionpatterns. As a result, the physics associated with a nonuniformplacementof chain ends is lost in this approach. Moreover, Ohta & Kawasaki useda random phase approximation for the free energy functional that isrigorously valid only for very weak compositional inhomogeneities, eventhough the inhomogeneities that characterize the SSLare O(1) in volumefraction. In spite of this, Ohta & Kawasakiwere able to obtain predictionsfor the domain periods and phase diagram that are qualitatively similarto those of Helfand-Wasserman and Semenov. They have also been ableto reduce the calculation of the microdomainfree energies to a purelygeometrical problem.

Finally, we mention the SSL theory of Anderson & Thomas(49), whichis the only theory thus far to contend with the OBDD hase. Theseauthors modified the Ohta-Kawasakiapproach to treat (A-B), star blockcopolymers. Although they found good agreement of the theoreticalOBDDattice parameters with experiment, Anderson & Thomas werenot able to predict thermodynamic stability of the OBDDhase in thecomposition windowwhere it is experimentally observed. Treatment ofthe OBDD hase with the methods of Helfand-Wasserman or Semenovhas yet to be carried out.WEAK SEGREGATION LIMIT (WSL)TheoryWhereas he theoretical advances in the SSLwere catalyzed by pioneeringexperiments involving the synthesis and characterization of modelcopoly-mers, developments in thc WSLwere strongly influenced by a seminaltheoretical paper by Leibler (4). Leibler considered the case of a mono-disperse A-B diblock copolymer melt with degree of polymerization N,


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538 BATES & FREDRICKSONcomposition f, and equal monomer volumes and statistical segmentlengths. For such a system, Leibler constructed a Landau expansion ofthe free energy to fourth order in a compositional order parameter field,~(r) = (6~bA(r)), where6~bA(r)= q~A(r) --f is the fluctuation in microscopicvolume fraction of A monomers at position r. Such expansions proveuseful in the vicinity of a second-orderor weak irst-order phase transition,where the amplitude of the order parameter field remains small in the lowtemperature phase (50). Indeed, the ODT f symmetric (f = 0.5) or weaklyasymmetric block copolymers is such a transition. Leiblers calculationwas particularly remarkable, because he provided microscopic expressionsfor the (Landau) expansion coefficients as functions of the incompatibility,~;N, and the copolymer composition. These coefficients were computedbymeans of the random phase approximation, introduced for polymer meltapplications by de Gennes 3, 51).By retaining only the leading harmonics in a Fourier representation ofthe various ordered-phase composition patterns, Leibler was able toexploit the Landau expansion and mapout the phase diagram ofa diblockcopolymer melt near the ODT.(This approximation of neglecting higherharmonics in the description of the composition patterns can be rigorouslyjustified only for the case of a second-orderphase transition, but is expectedto remain quantitative in weakfirst-order situations. Note also that theOBDDhase was not explicitly included in the free energy competition.)The phase diagram so obtained, in the parameter space of zN and f,is shown in Figure 6a. The Landau theory predicts a critical point at(~N)c = 10.5,fo = 0.5, where a compositionally symmetric diblock meltexpected to undergo a second-order phase transition from the disorderedto the lamellar phase. At such a transition, the amplitude of the lamellarpattern grows continuously from zero on lowering the temperature (i.e.increasing zN). The lattice constant (period) of the lamellar phasepredicted to be D ~ 3.23Rg ~ N1/2 at the symmetric ODT,consistent withthe WSL ssumption that the copolymcrs are only weakly perturbed by theinhomogeneous composition field. For asymmetric diblock copolymers,f :~ 0.5, the Landau heory predicts a weakfirst-order transition from thedisordered phase to the BCC pherical phase. In contrast to the situationin the SSL, it is important to note that the Landau heory predicts first-order transitions between solid phases that can be accessed by changingtemperature.Besides the phase diagram shown n Figure 6a, Leibler (4) providedexpression for the disordered phase structure factor, S(q) = (6(aA(q)6~A(q)), given

S (q) = N-F(x,f)-2)~ 1.


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BLOCK COPOLYMER THERMODYNAMICS 539(o)

18;~ LAMHex

14 -

Disordered~.,/~00 ~ ~ I I0.2 0.4 0.6 0.8

22

18

14

10

(b)

LAM

Disordered ~ = 10l I I I0.2 0.4 0.6 0.8Figure 6 Theoretical phase diagrams for diblock copolymers in the weak segregation limit:(a) mean-field theory (4); (b) fluctuation theory with N = 104 (62). LAM,Hex,correspond to lamellar, hexagonal (cylindrical morphology), and body-centered-cubic(spherical morphology) ymmetries, and the dashed curve represents the mean-field (Landau)stability limit. (Reproduced rom Ref. 63.)

2 2where x -- q Rg and F(x,f) is a dimensionless function of wavenumberand composition that is related to certain (Debye) correlation functionsof a Gaussian diblock copolymer (4). The most characteristic featurethis expression is the prediction ofa Lorentzian peak at x = x*(f) O(1),where F is minimum;he peak intensity diverges at the classical spinodalgiven by the condition F(x*,f)-2(%N)s = 0. In the Landau theory, thespinodal and critical point coincide for f= 0.5. It should be noted thatexpressions similar to Eq. 1 were also derived by LeGrand & LeGrand(52) and by de Gennes(51).The above expression for S(q) has facilitated the interpretation ofnumerousX-ray and neutron scattering measurementson partially labeledmodel diblock copolymers. Similar expressions for more complex blockcopolymer architectures, such as graft and star copolymers, have beenderived by Olvcra de la Cruz & Sanchez (53) and by Benoit & Had-ziioannou (54). Other workers (55-58) have extended the Leiblerexpression to incorporate polydispersity effects, which are often importantin practical applications. The latter extension is generally performed byaveraging the Debyecorrelation functions that constitute F(x,f) with anappropriate molecular weight distribution function, e.g. the Schultz-Zimmfunction.


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540 BATES & FREDRICKSONIt was recognized by Leibler (4) that the Landau heory, which predictsmean-field critical behavior (59), is inadequate in the vicinity of the f= 0.5critical point discussed above. Brazovskii (60) had previously demon-

strated that such critical points, predicted by mean-field theories for sys-tems exhibiting transitions between isotropic and striped (i.e. lamellar)phases, are suppressed by large-amplitude order parameter fluctuations.By means of a self-consistent Hartree approximation, Brazovskii showedthat the mean-field critical point is replaced by a weakfirst-order phasetransition, induced by fluctuations. It should be emphasized that such

fluctuation-inducedfirst-order phase transitions (61) have been predictedto occur in a variety of other physical systems, such as liquid crystalsand driven (nonequilibrium) fluids. Experiments to test these predictions,however, have been quite limited.Fredrickson & Helfand (62) extended Brazovskiis Hartree methodanalysis to the A-B diblock copolymer melt considered by Leibler. Theyfound that the fluctuation corrections are controlled by a Ginzburg pa-rameter (59) _~, proportional to the copolymer molecular weight, definedby _~= 63(R~3pc)2, where Pc is the number density of copolymersthe melt. For fixed incompatibility ~N, but N~ ~v, Fredrickson &Helfand found that Leiblers mean-field predictions are asymptoticallyapproached. For finite .~, however, the fluctuation corrections imposeboth qualitative and quantitative changes in the phase diagram (Figure6b) and scattering behavior. In particular, the Hartree approximationleads to a suppression of the symmetric critical point at (gN)c = 10.5,which is replaced by a weakfirst-order transition at (a lower temperature)(ZN)oDT 10.5 + 41.0/~- ~/3. The amplitude of the lamellar compositionpattern is predicted to be O()~r- ~/6) at the ODT.Because ~is of order 10-10~ for the typical experimental sample, these fluctuation corrections canbe substantial. The changes in the phase diagram for asymmetric diblocksare even more dramatic, as is indicated in Figure 6b for ~-= 10~. Animportant distinction with the mean-field diagram 6a is that the lamellarand the hexagonal phases are accessible at the ODTn the Hartree approxi-mation for f~ 0.5. However, the mean-field prediction of first-ordertransitions between ordered phases that can be accessed by changingtemperature is preserved in the Hartree approximation.The fluctuations manifest in the Fredrickson-Helfand theory also impactthe structure factor of the disordered and ordered phases. For the dis-ordered phase, the Hartree approximation for S(q) has the same wave-number dependence as in Eq. 1, but the bare Flory parameter g is renor-malized by composition fluctuations to an effective interaction parametergofr- This renormalized parameter depends on temperature, composition,and molecular weight and is related to the bare parameter by


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BLOCK COPOLYMER THERMODYNAMICS 541Zo~rN = ~N-- ~ [F(x*,f) -- 2~~rN] I]22N

where C(f) is a composition-dependent coefficient. For a symmetric melt,the peak intensity S(q*) attains a maximumalue that is O(N~~/3) at theODT. n contrast, the mean-field theory gives rise to a divergent peakintensity at the symmetric ODT.The Hartree approximation also predictsan isotropic scattering component in addition to the Bragg peaks) forthe weakly ordered phases (62, 63). This fluctuation component s also0(N~1/3), but is lower in intensity than the pretransitional disordered-phase component.

The Hartree approximation leads to an interesting physical picture of asymmetric diblock copolymermelt in the vicinity of the ODT63). Whereasthe Landau theory gives statistical weight only to the extremum com-position field configurations in the ordered and disordered phases, namelythe uniform and perfectly ordered configurations shown n Figure 7a, theHartree approximation also weights configurations like those shown inFigure 7b. The latter configurations have superimposed upon the extre-mum onfigurations isotropic composition fluctuations that have a pre-ferred wavelength, 2n/q*, but random directions and phases. The Fred-rickson-Helfand theory (62) suggests that the root-mean-squaredamplitude of these fluctuations is O(~-- 1/6) and is thus comparable o theamplitude of the long-range-ordered lamellar component. t is interestingto note that the typical equilibrium composition field configurations in adisordered diblock melt (which fluctuate in time) are reminiscent of thetransient nonequilibrium patterns encountered during the intermediateand late stages of spinodal decomposition (64).A recent theoretical study of Semenov65) suggests that the asymmetricwings of the phase diagram, i.e. f


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542 BATES CG FREDRICKSON

Figure 7 Instantaneous real-space composition patterns in the weak segregation limit: (a)mean-field theory, and (b ) fluctuation theory. +A versus r depicts the expected time depen-dence (f , # t 2 )of each morphology, where dA s the local volume fraction of A segments.Recent SANS results support the fluctuation picture near the order-disorder transition (seeFigure 8). (Reproduced from Ref. 63.)


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BLOCK COPOLYMER THERMODYNAMICS 543copolymers. Subsequent first order structural transitions into hexagonaland bcc phases are also predicted by the theory.At the time of this writing, the Hartree approximationhas been exploredfor triblock copolymers (66) but not for more complex block copolymerarchitectures, such as stars. However, he extension to concentrated andsemidilute diblock copolymersolutions with a neutral (nonselective) sol-vent has recently been carried out (67, 68). It has frequently been assumed(69) that copolymer solutions have a uniform distribution of the non-selective solvent in the ordered microphases. This suggests that in theconcentrated regime, where swelling effects are absent, the phase diagramofa copolymer olution is simply obtained by rescaling 2 to ~b2 in the meltphase diagram, where ~b is the copolymer volume fraction. Fredrickson &Leibler (67) have shown hat this "dilution approximation" method (69,70) neglects several aspects of the physics of such solutions. In particular,these authors demonstrated that even in the WSLhere is a tendency fora neutral, good solvent to accumulate at the interfaces of the micro-domains. This nonunilbrm placement of the solvent screens the unfavor-able A-B monomercontacts, but does so with a translational entropyprice. Screening occurs until that entropy cost exactly compensates theloss of contact enthalpy. For a good solvent, this compensation producesa periodic solvent composition profile with an amplitude that is N-~smaller than the amplitude of the A-Bcomposition profile. As the solventquality is decreased, the two order parameter fields have comparableamplitudes in the weakly-ordered microphases and a tricritical point isencountered (67). Another aspect of neutral copolymer solutions thatdistinguishes them from molten copotymers, is that the ODTs associatedwith a two-phase region in which disordered solvent-rich and orderedsolvent-poor phases coexist. This region is very narrow for goodsolvents,but broadens as the solvent quality is decreased. Finally, we note that byreformulating the melt Hartree approximation in terms of concentrationblobs, the important case of semidilute neutral copolymer solutions canbe treated (67, 68).ExperimentAs discussed in the previous section, bulk block copolymerscan be broughtinto the weak segregation regime by decreasing either Z or N. The formeris generally accomplishedby selecting structurally similar monomers. orexample, block copolymers (f ~ 1/2) prepared from styrene and a-methyl-styrene remain disordered (i.e. homogeneous) t around 180C for Mw5" 105 g/mole (71-73). In contrast, symmetric polystyrene-polydiene blockcopolymers exhibit an order-disorder transition at about this temperaturewhen 104 ~< Mw < 2" 104 g/mole, depending upon the polydiene type (e.g.


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544 BATES & FREDRICKSOINpolybutadiene or polyisoprene) and microstructure (74, 75). Since thesefirst reported, and in practice limiting cases, only a handful of blockcopolymers have been investigated in the WSL.

Pioneering studies by Cohen and co-workers (76, 78) on polydiene-polydiene diblock copolymers were conducted at about the time that thefirst WSLheory was developed. These investigations demonstrated WSLbehavior in 1,4-polyisoprene-l,4-polybutadiene (76, 77) and 1,4-poly-butadiene-1,2-polybutadiene (78) block copolymers that qualitatively sup-ported Leiblers mean-field predictions, and gave impetus for subsequentresearch based on this class of materials. Since then, WSLnvestigationshave been conducted with hydrogenated polystyrene-polydiene (75), poly-styrene-poly(para-methylstyrene) (79), poly(ethylene-propylene)-poly-(ethylethylene) (63, 80-82), and polystyrene-polymethylmethacrylate83, 84) block copolymers. In addition, modest amounts of neutral solvents(~< 60%) have been added to polystyrene-polydiene materials in orderdecrease the order-disorder transition-temperature, thus bringing the WSLinto the experimentally accessible temperature range (69, 85-87).

Within the weak segregation regime the most significant feature is theorder-disorder transition. Identification of the ODTemperature, denotedTODT,s often complicated by the weakfirst-order character of this phasetransition and the presence of significant composition fluctuations aboveand below TODT-Earlier studies established phase behavior based on thecalorimetric or dynamicmechanical evaluation of the glass transition; asingle glass transition is indicative of homogeneity i.e. disorder), whereastwo glass transitions signal microphaseseparation (i.e. order) (71-73,78). Although these techniques remain useful screening methods (80, 88),they are incapable of quantitatively establishing TODT.The ordering of a block copolymer is accompanied by gross changes inthe low frequency rheological properties as first shown by Chung et al(89a,b), and Pico & Williams (90) for a poly(styrene-butadiene-styrene)(SBS) triblock copolymer, and plasticized SBS, respectively. This behavioris characterized by the transition from a terminal dynamic mechanicalresponse for T > ToDT [e.g. G" ~ ~02 and G" ~ e~ for ~o ~ 0 where G" andG" are the dynamicelastic and loss moduli (91), respectively] to a non-terminal response for T < TODT-At sufficiently low frequencies, G, andto a lesser extent G", drop discontinuously as the temperature is raisedthrough the first-order ODT.This discontinuity provides a quantitativemeansof identifying ToDT;ypical rheometer temperature control affords ap-proximately IC precision in the determination of ToDT.This technique hasbeen demonstrated and exploited by several research groups studying bothdiblock (82, 92) and triblock (93-97) copolymers, and in our judgementrepresents the most eff~cient and accurate method or establishing ToDT.


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BLOCK COPOLYMER TIrIERMODYNAMICS 545In addition to these abrupt changes n the limiting low frequency(~o -~ 0)rheological properties at the ODT, omposition fluctuations near the phasetransition (see Figure 7) lead to significant departures from thermo-rheological simplicity (82, 92). A complete discussion of this dynamicalbehavior falls outside the scope of this review of block copolymer hermo-dynamics. Nevertheless, the continuous development of thermorheologicalcomplexity, particularly in the disordered state, is a direct manifestationof the fluctuation (i.e. thermodynamic)effects discussed previously and

described below, and should not be confused with the discontinuouschanges that mark the ODT 93).As with the strong segregation limit, SAXSnd SANSre very powerfuland important experimental tools for investigating block copolymers inthe weaksegregation limit. The choice of X-rays or neutrons as incidentradiation is dictated primarily by the choice of polymers, which determinethe contrast factor. Nonpolar systems governed by relatively large ~ par-ameters such as polystyrene-polydiene generally exhibit a sizeable electrondensity difference between components, which provides good X-ray con-trast. Accordingly, these materials are frequently studied by SAXS.Increasing block comparability by selecting structurally similar polymerssuch as isomers (78, 88) greatly reduces or eliminates X-ray contrast,making he use of SAXSither difficult or impossible. In this situation,deuterium labeling (e.g. deuterating one block) provides strong neutroncontrast, thus making SANS he experimental method of choice. Theseconsiderations are particularly important in the WSL.An intrinsicallyweakcontrast factor [i.e. similar pure componentelectron (for SAXS)neutron scattering length (for SANS) ensities] will be sensitive to smallchanges in the local specific volume. Therefore, any inhomogeneouslydistributed (i.e. composition dependent) excess volume of mixing willmodify this factor. If the local composition pattern changes with tem-perature, the contrast factor will vary with temperature. This effect willbe most severe when the composition profile is most temperature depen-dent, i.e. near the ODT.Such a spurious temperature dependence tothe scattering intensity would preclude the quantitative evaluation oftheory.Shortly after publication of Leiblers landmark WSLheory, Roe et al(25) demonstrated the existence of a broad temperature dependent SAXSpeak in a polystyrene-polybutadiene diblock copolymer above Toby. This,and other similar experiments (69), were found to be qualitatively con-sistent with Eq. 1, i.e. decreasing temperature in the disordered state ledto an increase in the peak scattering intensity. Quantitative assessments ofEq. 1 were first reported by Bates (56, 98) based on SANSmeasurementspartially deuterated 1,4-polybutadiene-l,2-polybutadiene (1,4PB-1,2PB)


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546 BATES& FREDRICKSONdiblock copolymers and by Mori et al (74), who studied polystyrene-polyisoprene (PS-PI) diblock copolymers by SAXS.These investigationsdemonstrated consistency between the measured and predicted disorderedstate structure factor forf~ 1/2, and for the first time provided estimatesof z(T) obtained by fitting Eq. 1 to temperature dependent small anglescattering data, as originally proposed by Leibler (4). Although the scat-tering results for the symmetric diblock copolymers produced results thatwere in good agreement with theory, Bates & Hartney (56) found a sig-nificant discrepancy between Eq. 1 and SANS ata obtained from a seriesof asymmetric (f ~ 0.25) disordered 1,4PB- 1,2PB samples. Along with theexpected scattering peak at wavevector q* these materials produced asignificant forward scattering component hat increased in intensity withincreasing N. Bates & Hartney (56) speculated that this feature couldderive from (domain-like) entities present in the disordered melt. Therecent prediction by Semenov(65) regarding micelle formation in thedisordered state (see previous section) is consistent with these observations.These measurements have not been confirmed, however, and the issue ofmicelle formation in the disordered state remains unresolved.As discussed in the previous section~ Fredrickson & Helfand (62) haverecently incorporated fluctuation effects into Leiblers original weakseg-regation mean-field theory, arriving at the phase diagram illustrated inFigure 6 (N = 104). Several experimentally testable differences betweenthe mean-field and fluctuation theories can be identified. As the ODTsapproached in the disordered phase, both theories anticipate a rapidincrease in the peak scattering intensity I(q*) ,~ S(q*) as indicated by Eq.1. However,for the general case in which ~ = ~T-~+ fl (see Introduction)the mean-field and fluctuation theories differ significantly in the predictedtemperature dependence f I(q*); the former predicts I- l(q*) to be linearT- 1, whereas fluctuation effects produce a nonlinear relationship betweenthese parameters (see Eq. 2).

Beginning with Roe and co-workers (25), the order-disorder transitionhas been examined n a variety of polystyrene-polydiene diblock (25, 76,86), triblock (95), and star-block (86, 87) copolymers and hydrogenatedpolystyrene-polybutadiene diblock copolymers (75) by small-angle X-rayscattering. These studies have relied on SAXS ata obtained as a functionof temperature for determining TooT. In general, the ODT as been cor-related with the temperature when a deviation from lincarity in a plot ofI ~(q*) versus T ~ is observed, which assumesmean-field behavior (25, 75,95). Alternatively, Hashimoto t al (86, 87) have relied on the temperaturedependenceof the scattering peak position q* in fixing TooT. [Previouslythese authors reported q* ~ TO for T > TODT and q* ~ T1/3 for T < TODT(69). Neglecting the intrinsic polymer coil thermal expansivity, the WSL


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BLOCK COPOLYMER THERMODYNAMICS 547and SSL heories predict q* ~ TOand q* ~ T1/6, respectively (see Theorysections).] In all these SAXSstudies the ODT ppears as a continuoustransition as evidenced by an unbroken (q*, T), contrary to the predictionof a first-order transition by both mean-field (f:~ 1/2) and fluctuationtheory. None of these publications corroborate the assignment of ToDTwith rheologieal evidence of the phase transition.Recently, Bates and co-workers (80) reported the preparation of fullysaturated hydrocarbon diblock copolymers in the WSL.A series of mono-disperse, f~ 0.55, poly(ethylene-propylene)-poly(ethylethylene) (PEP-PEE)samples were studied rheologically (82) and by SANS63, 81).to the chemical similarity between blocks, this system exhibits an ODT troughly five times the molecular weight of an equal composition poly-styrene-polydiene material. For example, for Mw 57,400 g/mole, Bateset al (63, 80-82) find TOD~ 100C. This higher molecular weight bringsthese polymerswell into the rheologically entangled state at the ODT,whichfacilitates determining TODT82). In principle, higher molecular weightpolymersare also better candidates for evaluating the statistical mechan-ical WSLheories that are premised on a large N. Bates et al (63) haveshown an exact correspondence between the temperature at which therheological properties in a PEP-PEE ample are discontinuous and whereI(q*) exhibits a subtle (20%) discontinuity. These results conclusivelydemonstrate the first-order character of the ODT nd underscore the valueof dynamic mechanical analysis in establishing TooT. However, contraryto Hashimotoet al (86, 87), Bates and co-workers (63) report q*(T)unaffected by the ODT.

A full evaluation of the PEP-PEE ANSesults revealed the first clearevidence of composition fluctuations near the ODT.In the disorderedstate, the principle scattering reflection could be quantitatively fit with thetheoretical structure factor (Eq. 1). In addition, a shoulder was apparentat q ~ 2q* that became more prominent as the temperature was loweredtowards TODT.This feature is not accounted for by current theory andsuggests that the disordered phase maypossess more"structure" (i.e. largecomposition gradients) than has previously been assumed. Overall, thedisordered state SANS tructure factor closely resembles the structurefactor characterizing the final stage of spinodal-decomposition in a sym-metric binary polymer mixture (99), which is the basis for the real-spacemorphologyof the fluctuating disordered phase depicted in Figure 7. Asshown in Figure 8, I-l(q,) is clearly nonlinear in T-~ over the entiretemperature range examined, which extends 56C above Tooa-. A quan-titative comparisonof the mean-field and fluctuation theory predictions isalso shown n Figure 8. [These calculations were madewithout adjustableparameters (63); N, fand z(T) were determined independently (82).]


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548 BATES & FREDRICKSON

5

4

DISORDERED

00

0

MEAN-FIELD

00

00

2.t 2.2 2.3 2.4 2.5 2.6Fiyure 8 Reciprocal SANS eak intensity versus inverse temperature for a disordered modelPEP-PEE iblock copolymer near TooT. he mean-field and fluctuation curves have beencalculated (no adjustable parameters) by using the Landau(4) and fluctuation (62) theories,respectively, as described in Ref. (63).

comparison onfirms the predicted significance of fluctuation effects in thedisordered state near the ODT nd rules out the use of a mean-fieldassumption in evaluating I(q*, T) in these regions of the block copolymerphase diagram.Bates et al (63) also investigated the ordered state of a PEP-PEEpeci-menby using SANS. n order to facilitate these measurements, a samplewas shear-oriented based on the principles established by Mathis et al(100). Scattering experiments revealed a lamellar morphology hat per-sisted up to TooT, confirming the fluctuation theory prediction that for


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BLOCK COPOLYMER THERMODYNAMICS 549slightly asymmetric compositions (here f= 0.55) the lamellar orderedphase should lead directly to the disordered phase (see Figure 6b); mean-field theory predicts a lamellar-hexagonal-(body-centered cubic)-dis-ordered sequence of phase transitions (Figure 6a). Fluctuations were alsoevident in the two-dimensional SANS attern (obtained from the orien-ted specimen) as the temperature approached TODT, n agreement withtheory. Overall, the fluctuation theory is remarkably consistent with the(limited) experimental data (PEP-PEE, f = 0.55) available for compari-son at the time of this writing, thus leading us to speculate that thereal-space morphologies near the ODTesemble those depicted in Figure7b; for T>> TOD~ nd T


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550 BATES & FREDRICKSONSURFACE BEHAVIORExperimentBlock copolymersurface properties have been a topic of great interest forseveral decades due in part to applications ranging from the formulationof adhesives to lubricating surfaces. Until recently, surface characterizationtools have been limited to indirect methods such as X-ray photoelectronspectroscopy (XPS) and contact-angle wetting experiments. The past fewyears have witnessed vigorous growth in the area of surface analysis,particularly regarding block copolymer surfaces, due to the developmentof several new direct quantitative techniques, most notably dynamicsecondary ion mass spectroscopy (SIMS), and neutron reflectometry.

Early investigations of the wetting properties of undiluted block copoly-mers, which dealt mainly with semicrystalline materials, indicated a sig-nificant degree of surface enrichment in one block component 102). Thisphenomenon was particularly evident in block copolymers containingpolydimethylsiloxane, which exhibited low energy surfaces, consistent withthe surface properties of polydimethylsiloxone homopolymer. Surfaceenrichment of polystyrene in polystyrene-poly(ethylene oxide) diblockcopolymers was also quantified by Thomas & OMalley (103) using XPSmeasurements. Recently, Green et al (104) have employedXPS o evaluatethe surface properties of a series of polystyrene-polymethylmethacrylatediblock copolymers. They report a molecular weight dependent surfacepreference for polystyrene. Although these studies clearly demonstrate theexistence of a preferred surface component, which is easily rationalizedbased on the pure component surface-tensions, the analytical methodsdiscussed thus far are incapable of quantitatively delineating the actualsurface profile or topology.Direct evidence of surface segregation in a polystyrene-polyisoprene di-block copolymer was obtained with TEM y Hasegawa& Hashimoto (105).Solvent cast specimens were stained with osmium-tetraoxide, embeddedin epoxy, and ultramicrotomed in preparation for microscopic examination.Regions of the ordered material (SSL) were photographed with lamellaearranged perpendicular and parallel to the free surface. In both cases, athin polyisoprene layer, approximately half a bulk lamellar dimensionthick, existed at the polymer urface. This technique is suitable for observ-ing strongly segregated surfaces but is not likely to be effective in the weaksegregation regime where staining could influence the system morphology.SIMS s an alternative direct profiling technique that has recently beenapplied to the investigation of block copolymer surfaces. This methodrelies on the intensity of secondary ons (e.g. ~H+, 2H+, 12C+, etc.) emittedas a function of time during primary ion beamsputtering of the surface. A


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BLOCK COPOLYMER THERMODYNAMICS 551detailed SIMSanalysis of a polystyrene-polymethyl methacrylate diblockcopolymer ilm by Coulonet al (106) illustrated howeffectively this methodprovides images of periodic lamellar structures that were formedparallel tothe block copolymer ree surface upon annealing at elevated temperatures.These authors estimate an instrumcnt resolution of about 125 ~. Anotherpopular ion beam technique that finds application in polymer surfacestudies, forward recoil spectroscopy (FRS), is generally not appropriatefor block copolymer urface problemsbecause of current resolution limita-tions (>~200~) (107).The most recently developed surface analytical technique with directapplicability to block copolymers is neutron reflectometry. Here a col-limated neutron beam produced by either a rcactor or pulsed neutronsource is directed at a samplesurface and the reflected intensity is measuredas a function of incident angle. Above he critical angle for total reflection(Oc ~ 0.1), the reflected intensity is attenuated due to interference effectscreated by the sample composition profile, the free surface interface, andthe substrate interface. By modeling the reflection curve, a detailed com-position profile can be developed, with a high degree of sensitivity to bothspatial dimensions (~ 2 ~ resolution) and form. Russell and co-workers(83, 108, 109) have published a series of elegant neutron reflection studiesconducted with as-cast and annealed polystyrene-polymethylmethacrylatediblock copolymers. A representative neutron reflection curve for anannealed specimen s shown n Figure 9. The solid curve in this illustrationis the best-fit calculated reflectivity and corresponds to the compositionprofile shown n the inset. A slight variation in the composition profile(inset) produces significant changes n the calculated reflectivity as illus-trated by the dashed curves. Russell and co-workers have exploited thispowerful technique in examining a number of issues relating to surfacesegregation in thick and thin block copolymer films, including surfaceordering and disordering as a function of temperature (109), and thequantitative evaluation of the interracial profile in ordered lamellae (108).They report close agreement with mean-field theory (110) (see followingsection) for the surface induced formation of structure near the air-polymerinterface, and a coincidence between the surface and bulk ODT em-perature (109). Quantitative agreement with mean-field theory for theserelatively low molecular weight specimens (N ~ 300) in the bulk state andat the surface is surprising given the strong fluctuation effects found inhigher molecular weight PEP-PEE amples (63, 81, 82) (see FigureTheoryWeare only aware of one theoretical study that has explicitly consideredthe behavior of block copolymermelts near a free surface or solid substrate.


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552 BATES & FREDRICKSON

10-1 - -10-2 -

R - -~1>.0103 ~ ~,,~.,o 4~o 8~o 12~ooeoo-104 ~o~ ~_

105I I I I I I I0-6.006 0.02 0.04 0.06 0.08

k0lA-~)ig#re ~ Neutron reflecfivJty R as a uncfion o[ the neutron momentumperpendicularo the suffacco a thin annealed o]ydeuterostyrene-po]ymethy]metSacrylatedib]ockcopo]ymerilm. Thesofid aadd~s~ed #~esare the calculatedcompositionrofiles shown n the i~set (~/~ is the neutron cattefin8 lensth density) asreportedn R. (108). (Provided y T.

Fredrickson & Helfand (111) treated the case of a symmetric diblockcopolymer melt in the WSLand in contact with such a surface. Heemployed a Landau approach, adding a surface frcc energy contributionto Leiblers bulk free energy expression (4). The surface term contained twophenomenologicalparameters Hi and a l that describe the modifications ofthe (A-B exchange) chemical potential and the interaction parameterthe copolymer layer adjacent to the surface, respectively. By minimizingthe sum of these bulk and surface frcc cnergy contributions with rcspcct tovariations in the composition field employed y Leibler, ~O(r), Fredricksonderived an Euler-Lagrange equation for the composition pattern in thepresence of a surface. At temperatures above the bulk ODT,zN < (ZN)oDT, any slight preference of monomerA or B for the surface,]H~[ > 0, proves sufficient to induce sinusoidal compositional order intothe melt. The wavenumber f these oscillations is roughly equal to q*,which characterizes the peak position in Leiblers diblock structure factor.Moreover, the oscillations are exponentially damped on the scale of


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BLOCK COPOLYMER THERMODYNAMICS 553the disordered phase bulk correlation length, which is singular near thespinodal in Landau theory, 4+ ~R~(1--X/)~s) -1/2. Similarly, belowthe bulk ODT,Fredrickson finds that the amplitude of the lamellarcomposition profile is modulated out to distances from the surface oforder the correlation length, 3-~ Rg(X/Xs-1) -1/2. Various surfacephase transitions (110) are also possible for H1 = 0, depending on thesign of a 1, although such conditions are likely very difficult to achieve inthe laboratory.In spite of the good agreement between the mean-field surface com-position profile and the experiments of Russell and co-workers (83, 108,109), we expect that extensions of the above theory to incorporate fluc-tuation effects will prove necessary. Extensions o treat systems with cylin-drical, spherical, or OBDDulk order would also be valuable.DISCUSSION AND OUTLOOKA survey of recent developments in block copolymer thermodynamics ispresented in the previous sections. Wehave not attempted an exhaustivereview of this expansive subject. Instead, we have restricted our attentionto studies dealing with fundamental ssues pertaining to order and disorderin this complex and fascinating class of materials. Recent theoreticaladvances in this area relate exclusively to model(A-B), type materials; forthe most advancedstudies involving fluctuation effects, the focus narrowsto A-B diblock and triblock copolymers. In considering a vast experi-mental literature we have opted to limit our scope to selected publicationsrelevant to current theory, or those likely to provoke new theoreticalefforts. Despite these restrictions, there still exist a number f significantdeficiencies in our understandingof this field.At present there is no theory capable of predicting the existence of theordered bicontinuous double diamond phase in the strong segregationlimit. In addition, there is no experimentalor theoretical evidence to proveor disprove that the OBDDhase can be accessed at the ODT.However,the beautiful morphological studies of Thomasand co-workers, and theconnections that they have made to the mathematical science of minimalsurfaces, should provide new impetus for exploring fresh approaches tothe above problems.In our opinion the major unresolved issues in block copolymer thermo-dynamicsare associated with the order-disorder transition, which we haveincluded under the heading of weak segregation limit. Foremost amongstthese problems is the notion of the WSLtself. The WSL ssumptionoriginated with Leiblers mean-field treatment (4), and was retained


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554 BATES & FREDRICKSONFredrickson & Helfand (62) when fluctuation effects were incorporatedinto the theory. Our binary classification schemeof WSL nd SSL remainsa convenient and obvious one when dealing with currently available theor-ies. However, categorizing the experimental studies near the ODT nderthese two headings is not so straightforward. Clearly there exist limits atsufficiently large and small values of zN at which block copolymers con-form to the SSL and WSL ssumptions. Nevertheless it must be recognizedthat at present there is no theoretical estimate of the size or location ofthe intermediate region between these limiting behaviors. WSLheoriesassume a sinusoidal composition profile in the ordered phase, and unper-turbed Gaussian coil behavior. Whether this accurately reflects the truesituation in the transition region is simply unknown.An important chal-lenge for theorists is to develop a comprehensive heory that deals withthe crossover from the WSLo the SSL.Although recent SANSxperiments (63, 81) support the qualitative (andto some extent, quantitative) predictions regarding composition fluc-tuations near the ODT, everal troubling inconsistencies challenge theunderlying WSL ssumption. The observation of a shoulder at approxi-mately twice the principle SANSeflection at temperatures above TODTyBates et al (63) cannot be explained by the perturbative methods mplicitin the WSLheories. In fact, similarities between the disordered statestructure factor and the final stage spinodal decomposition scatteringpattern suggests a rather strongly segregated, albeit disordered, state. Thisrecent finding is reminiscent of the SAXSesults reported by Roe et al(25) nearly a decade ago, in which narrow (~ 20/~) interfacial thicknesseswere determined above ToDr. Stronger than predicted segregation in thedisordered state might also account for the anomalous low frequencyrheological response found near the ODT 82, 92, 111). Off-symmetrySANSmeasurements also suggest the existence of "domain-like" entitieswithin the disordered melt (56), which have been predicted by Semenov(65) in the limit of small (or large)These experimental findings near the order-disorder transition raiseserious questions concerning the weak segregation limit assumption, andmay nvalidate our classification scheme. Webelieve that the resolution ofthis issue will require both new theoretical approaches and additionalquantitative experiments.Finally, we have included block copolymersurfaces in this review. Onlyvery recently have quantitative experimental techniques (e.g. SIMSandneutron reflectometry) been introduced in this field. Although identifyingsubjects that hold unusual promise is risky, we feel quite certain thatdevelopments n this area will come apidly, and will have wide scientificand technological impact.


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BLOCK COPOLYMER THERMODYNAMICS 555ACKNOWLEDGMENTSWeare indebted to our collaborators E. Helfand, R. Larson, L. Leibler,and J. Rosedale for their contributions to our understanding of the issuescovered in this review. F.S.B. also acknowledges support from the NSFunder PYI Grant DMR-8957386.

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Frank S. Bates- Block Copolymer Thermodynamics: Theory and Experiment

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