Ordering of Sphere Forming SISO Tetrablock Terpolymers on aSimple Hexagonal LatticeJingwen Zhang,† Scott Sides,‡ and Frank S. Bates*,†

†Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, United States‡Tech-X Research Corporation, Boulder, Colorado 80303, United States

ABSTRACT: Hexagonally ordered spherical and cylindricalmorphologies (P6/mmm and P6/mm space group symmetries)have been identified in bulk poly(styrene-b-isoprene-b-styrene-b-ethylene oxide) (SISO) tetrablock terpolymers. Thesematerials were synthesized by adding up to 32% by volumeO blocks to a parent hydroxy-terminated SIS triblockcopolymer containing 40% S by volume, and the resultingmorphologies were characterized by small-angle X-rayscattering, transmission electron microscopy, differentialscanning calorimetry and dynamic mechanical spectroscopy.Disordered, spherical and cylindrical phases were documented with increasing O content, where both ordered states exhibithexagonal symmetry. Theoretical calculations based on a numerical self-consistent field theory for polymers provide crucialinsights into the molecular configurations associated with these morphologies. These results offer a new approach toindependently control domain shape and packing in block copolymer melts through manipulation of the magnitude andsequencing of the binary segment−segment interactions (χSI ≤ χSO ≪ χIO), which dictate core segregation and the effectiveinterdomain interactions.

I. INTRODUCTIONModern synthetic polymerization chemistry affords access to arapidly expanding portfolio of block and graft copolymerarchitectures generated from a rich array of monomers. Linkingthe resulting morphological features and related physicalproperties with the molecular structures constitutes a centralchallenge to polymer scientists and engineers. This articledescribes new and surprising results regarding the ordering ofsphere forming ABAC tetrablock terpolymers.Access to powerful characterization tools, most notably

synchrotron small-angle X-ray scattering (SAXS), and theapplication of sophisticated theoretical techniques have greatlyenhanced the pace of discovery in the field of blockcopolymers. Linear AB diblock copolymers, the simplest case,provide instructive examples of this synergy. Until the mid-1980s, just three diblock morphologies were recognized basedprimarily on data obtained using transmission electron micro-scopy (TEM) and laboratory based SAXS instruments: spheres,cylinders and lamellae. The concept of ordered bicontinuousphases, well established in the area of surfactancy,1 evolvedbetween 19852 and 1994,3 eventually resulting in theidentification of the fascinating gyroid phase. (Initial claims ofa cubic double diamond phase are now generally attributed tothe gyroid.4,5) Significantly, experiments and theory both playedcritical roles in unraveling the molecular factors responsiblefor formation of the gyroid.6 Several years ago self-consistent fieldtheory (SCFT) anticipated a fifth phase in diblocks,7 anorthorhombic network structure dubbed O70. (This remarkableprediction evolved from an investigation of ABC triblocks).8

Soon after publication of this theoretical work synchrotron SAXSexperiments revealed the existence of O70 within a narrow slice ofparameter space9 defined by the composition (f) andcombination parameter χABN, where χAB is the Flory−Hugginssegment−segment interaction parameter and N is the degree ofpolymerization. Recently we documented the σ-phase in a lowmolecular weight sphere forming poly(isoprene-b-lactide) (IL)diblock copolymer melt using synchrotron SAXS and TEM.10

This sixth ordered equilibrium structure, which has not beenaccounted for theoretically, makes contact with several seeminglyunrelated fields in materials science.10−12

Adding more block types geometrically expands thecomplexity of microphase formation. Linear ABC triblockterpolymers have produced over 30 well-defined orderedmorphologies and more are likely to be discovered. Threechemically distinct block types offer attractive design flexibility(e.g., glassy, rubbery, and semicrystalline domains arrangedin tailored microstructures)13−24 but entail a daunting array ofstructural choices. Fortunately, SCFT can provide pivotalguidance in relating morphology to the block sequence and thetrio of interaction parameters, χAB, χBC, χAC, as evidenced by thesuccessful modeling of entire ABC phase portraits.8,25,26

So, why add another block to an already complicated system?We believe that the ABAC tetrablock terpolymer architecture(and the coupled version ABACABA) affords the polymer

Received: September 29, 2011Revised: November 20, 2011

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scientist and engineer a rich set of new design options. Forexample, ABA triblocks make excellent thermoplastic elas-tomers and tough plastics. Addition of a fourth chemicallydistinct block can direct this mechanically robust unit intootherwise inaccessible morphologies depending on the relativemagnitude of χAB, χBC and χAC. Recently we have exploited thisidea with fully saturated CECPCEC multiblocks that are drivento order from the homogeneous melt by poly(ethylene) (E)crystallization.27 Combining minority amounts of glassypoly(cyclohexylethylene) (C) and E with a majority of rubberypoly(ethylenepropylene) (P) in a heptablock architecture leadsto a versatile new class of thermoplastic elastomers witha convenient and invariant (with respect to molecular weightand composition) processing temperature. These characteristicsresult from the combined effects of the specified blocksequence and χEP ≪ χCP < χCE.

28

In this article we examine the consequences of invertingthe block placement with respect to the interaction parametersin a series of SISO tetrablocks, where χSI ≤ χSO ≪ χIO. Thisarrangement discourages contact between O and I blocksleading to spherical domains of O, clad in a shell of S andembedded in a matrix of I and S. These tetrablocks have beenprepared by adding O blocks of varying length to a commonSIS parent triblock (40% S) that is disordered above about80 °C (i.e., T > Tg). We report a new finding that complimentsour recent discovery of the σ-phase in a similar material pre-pared from a compositionally symmetric SIS triblock (50% S):packing of spherical microdomains on a simple hexagonallattice (P6/mmm symmetry). SCFT calculations do not accountfor the experimental packing symmetry but do anticipate aspherical morphology (BCC symmetry) and provide valuableinsight into the distribution of blocks within the orderedstructure.These findings lead us to a conjecture that the tetrablock

terpolymer molecular architecture offers previously unrecog-nized potential for controlling the shape and packing symmetryof microphase separated block copolymers through delicatemanipulation of the effective interdomain interactions.

II. EXPERIMENTAL SECTIONSynthesis and Chemical Characterization. Figure 1 shows the

molecular architecture for the poly(styrene-b-isoprene-b-styrene-b-

ethylene oxide) (SISO) tetrablock terpolymers described in thisarticle, which were prepared using a two-step living anionicpolymerization procedure that is described in detail in a previouspublication.22 Monomers and solvent were purified as describedelsewhere.29 In the first step, styrene was initiated by sec-butyllithiumand polymerized at 40 °C in cyclohexane under argon. An aliquot ofthis product was removed for size-exclusion chromatography (SEC)analysis prior to the addition and complete reaction of isoprenemonomer, followed by another carefully metered amount of styrene,equal in mass to the first charge. After completion of this triblockpolymerization, a 10-fold molar excess of ethylene oxide, relative tostyryllithium, was added to the living polymer solution and allowed toreact for 60 min and then the reaction was terminated with excessdegassed methanol. The hydroxyl functionalized triblock copolymer

product (SIS−OH) was precipitated in methanol, decanted, and driedin a vacuum oven at room temperature. In the second step, theSIS−OH compound was reacted with potassium naphthenalide intetrahydrofuran at 40 °C followed by the addition of a measuredamount of ethylene oxide, then terminated with excess degassedmethanol after 48 h. Five SISO tetrablock copolymers were producedin this way, each containing the same SIS core and varying O blocklengths. Excess tetrahydrofuran solvent was removed by rotaryevaporation then the SISO polymers were redissolved in dicholoro-methane and washed once with an aqueous sodium bicarbonatemixture and several times with distilled water to remove residual salts.Finally, the tetrablocks were precipitated in a 3:1 solution of methanoland isopropanol, decanted, filtered, redissolved in benzene and freeze-dried under vacuum. Before freeze-drying, 0.5 wt % (relative to thepolymer) of BHT antioxidant was added to the solution to minimizesubsequent degradation of the PI blocks. Figure 1 illustrates themolecular structure of the SISO tetrablock terpolymers.Chemical Characterization. SISO tetrablock copolymer compo-

sitions were determined by 1H nuclear magnetic resonance (NMR)spectroscopy using a 300 MHz Varian instrument operated at roomtemperature with deuterated chloroform as the solvent. We estimatedthe mole fractions of styrene, isoprene, and ethylene oxide repeat unitsbased on the integrated area under well-defined resonances, and thesevalues were converted to weight fractions and volume fractions basedon the published homopolymer densities at 140 °C (ρS = 0.969 g/cm3,ρI = 0.830 g/cm3, ρO = 1.064 g/cm3).30

The absolute number-average molecular weight of the firstpoly(styrene) block (Mn,S) and the molecular weight distributionand polydispersity index (PDI) of all the specimens, were obtainedusing size exclusion chromatography (SEC). A Waters 150-C ALC/GPC configured with Phenomenex Phenogel columns was operatedwith chloroform as the solvent at room temperature. Polystyrenestandards from Polymer Laboratories were used as the basis to calculatethe polydispersity, which in all cases was less than 1.1. The overallnumber-average molecular weight of each tetrablock copolymer,Mn,SISO, was calculated using the Mn,S and the weight fractionsdetermined by 1H NMR. Table 1 lists the chemical characterizationdata for all the SISO samples.Small-Angle X-ray Scattering (SAXS). Synchrotron SAXS data

were obtained at the DuPont−Northwestern−Dow SynchrotronResearch Center at the Advanced Photon Source using sample−detector distances of 5.50 and 6.52 m with radiation wavelength λ =0.729 Å. Scattering patterns were recorded with a Mar CCD areadetector. Two-dimensional data were azimuthally averaged and arerepresented as intensity vs scattering wave vector, q = |q| = 4π/λsin(θ/2), where θ is the scattering angle. SIS and SISO-1 wereannealed at 110 °C for 5 min at Argonne before taking measurements.The other tetrablock specimens were annealed at 120 °C undervacuum for 1 day, quenched in liquid nitrogen, reheated to 120 °C atArgonne, and annealed for 5 min before taking measurements.Temperature was controlled to within ±1 °C using a DSC instrumentwhile specimens were maintained under a helium purge.Transmission Electron Microscopy (TEM). TEM data were

obtained using a Tecnai T12 transmission electron microscopeoperating at 120 kV at the University of Minnesota CharacterizationFacility. Samples were annealed under vacuum at target temperaturesfor specified times prior to quenching in liquid nitrogen to preservethe melt morphology. Thin slices (∼60 nm) of polymer specimenswere prepared by microtoming at −70 °C using a Reichartultramicrotome fitted with a Microstar diamond knife and collectedon copper grids (Ted Pella). Microtomed sections were stained withthe vapor from a 4% aqueous osmium tetraoxide solution for 10 minto obtain electron contrast; this reagent reacts preferentially with thePI blocks.Differential Scanning Calorimetry (DSC). DSC experiments

were conducted on a TA Instruments Q1000 DSC. Approximately5−10 mg of polymer was heated to 150 °C to minimize the effects ofprior thermal history before cooling to −100 at 10 °C/min. Datawere acquired during a second heating cycle, also at 10 °C/min. Thedegree of crystallinity of the PEO was determined by integrating the

Figure 1. Molecular structure of SISO tetrablock terpolymers.

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baseline-corrected melting peaks and normalizing this value to the heatof fusion of bulk PEO (213 J/g),31

(1)

Dynamic Mechanical Spectroscopy (DMS). DMS data wereobtained between 80 and 250 °C with a Rheometrics Scientific ARESstrain-controlled rheometer fitted with 25 mm diameter parallel plates;these temperatures are above the glass transition and meltingtemperatures of the PS and PEO blocks (Tg,S ≈ 60 °C, and Tm,O <60 °C). Samples were prepared by compression-molding the polymerpowder into 25.0 mm ×0.8 mm disks at 140 °C using 2000 psi ofpressure for 5 min. All samples were initially heated to 250 or 10 °Cabove the order−disorder transition temperatures (TODT), whicheverwas lower, to minimize the effects of prior thermal history. Aftercooling, isochronal (ω = 1 rad/s) dynamic elastic (G′) and loss (G″)moduli were recorded with a strain amplitude of 1% while heating at1.5 °C/min. A linear viscoelastic response was verified by runningstrain sweeps from 0.1 to 10% at representative temperatures.Isothermal frequency sweeps (100 ≤ ω ≤ 0.1 rad/s) also wereconducted at selected temperatures.

III. THEORETICAL CALCULATIONS

Numerical self-consistent field theory (SCFT) for densepolymer melts has been highly successful in describing complexmorphologies in block copolymers.32 Field-theoretic simula-tions such as these are able to access large length and timescales that are difficult or impossible for particle-basedsimulations such as molecular dynamics, while still incorporat-ing more realistic polymer models than many macroscopic,continuum simulations. We briefly describe the model andsimulation method for a dense melt of AB diblock copolymersin bulk. For the experimental system in this paper the A specieswill represent monomers of poly(styrene) and the B species willrepresent monomers of poly(isoprene); extension to SISO followsthe methodology outlined for an AB diblock. The chemicalspecificity of S and I blocks is captured in the model through thevalue of the Flory parameter χ, which controls the strength of thechain segregation that drives the formation of nanoscalemorphologies. The Hamiltonian of a dense melt of n polymerchains containing distinct monomers A and B can be written as:

(2)

The first term in H gives the free-energy contribution fromthe so-called “Gaussian thread” model.33 This model can bederived as the continuous limit of a polymer chain consisting of

spherical beads connected by harmonic springs (the radius ofgyration of an unperturbed chain in a dense melt of identicalchain is Rg0). The position r of a bead s contour steps along theαth chain is given by the space curve r(s). The second term inH represents the free-energy contribution from a model inwhich dissimilar monomers in contact have a higher free-energythan like monomers in contact. The strength of this tendencyfor chemically dissimilar monomers to segregate is para-metrized by the aforementioned χ value and is a function ofthe specific chemical species A and B as well as temperature.The model given by eq 2 is for a linear AB diblock copolymer.However, the model is easily generalized to a tetrablockcopolymer like those described in this study.The Hamiltonian represented by eq 2 can be used to form

the partition function of a dense melt of n copolymer chains.The monomer density operators may be recast as continuousmonomer density fields by application of a Hubbard-Stratonovich transformation34 that introduces chemical poten-tial fields ω(r) that are conjugate to the monomer densityfields. The partition function then takes the form:

(3)

where ρA and ρB are the smeared out density fields and ωA andωB are the conjugate chemical potential fields that areintroduced through the field theory transformations. The pfield is introduced through the constraint of overall monomerincompressibility. The integrals are over all possible realizationsof the fields with F given by:

(4)

where Q is the single-chain partition function,32 and is afunctional of the chemical potential fields and a function of thechain length N. The integrals in eq 3 are still intractable, so amean-field approximation is made where the extremum of F isassumed to make the largest contribution to the overall sum. Asystem of equations is then generated that needs to be solvedself-consistently in order to satisfy the mean-field approxima-tion. An iterative algorithm is used to find solutions for the ρ(r)and ω(r) fields that satisfy the SCFT mean-field equations.34

For the rest of this discussion, this SCFT algorithm is denotedby Rsc . For fields at iteration step n, the fields at iteration n+1can be obtained by ωn+1 = Rsc[ωn]. However, for large simula-tion grids this iterative algorithm can lead to field solutions thatcontain topological defects,35 analogous to the defects observedin experimental TEM images.

Table 1. SIS and SISO Characterization Data

sample Mn/kDa PDI f Sa f I

a f Oa phaseb d/nmc,d % cryst.e TODT/

oCf

SIS 23.3 1.03 0.39 0.61 0.00 DIS - 0.0 <80SISO-1 23.6 1.04 0.38 0.60 0.02 DIS - 36.2 <80SISO-2 24.4 1.04 0.35 0.56 0.09 HEX 21.8, 18.9 39.8 213 ± 2SISO-3 25.0 1.08 0.34 0.53 0.12 HEX 25.8, 22.3 55.3 >250SISO-4 26.1 1.06 0.32 0.50 0.19 HEX 29.7, 25.7 68.3 >250SISO-5 29.0 1.07 0.27 0.41 0.32 HEX 35.5 70.8 >250

aVolume fractions calculated with from densities at 140 °C.30 bHEX-hexagonally packed spheres for SISO-2, -3, -4 and hexagonally packed cylindersfor SISO-5. cTemperature is 120 °C. dLattice dimensions listed for SISO-2, -3, -4 correspond to a and c and characteristic morphological length scale(d = 2π/q*) for SISO-5. ePercent crystalline of the PEO domains was determined by differential scanning calorimetry. fOrder−disorder transitiontemperatures were determined by dynamic mechanical spectroscopy.

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A “spectral filtering”36 algorithm has been shown to helpreduce the presence of topological defects in numerical SCFTcalculations of block copolymer structure by removing certainfrequency components of the chemical potential fields duringthe relaxation of these fields toward the lowest free-energyconfiguration. The spectral filtering function Xsp is defined bythe following:

(5)

where the ω̂ represents a Fourier transformed field and fcutparametrizes the “strength” of the filtering function and takesvalues between 0 and 1. The SCFT mean-field algorithm Rsc isapplied for a sufficient number of iterations nstart to initiatephase segregation. Then the filtering algorithm Rfilter is applied

(6)

where the strength of the filtering fcut is a free parameter for aparticular simulation run. This spectral filtering has been foundto assist the relaxation algorithm Rsc in finding orderedmesophases. The spectral filtering essentially keeps the largestcomponents of the frequency spectrum of the transformedchemical potential field, which then acts as an effective initialcondition for the self-consistent algorithm. The assumption forthis procedure is that the topological defects one is trying toremove will tend to have the smallest frequency components.The filtering operation Rfilter is applied every 1000 fielditerations, i.e., applications of Rsc. This procedure continuesuntil an ordered mesophase is obtained or the fields reach asteady state configuration. This filtering procedure (alsodescribed elsewhere36) has been improved upon in this study.The older spectral filtering algorithm (outlined above) uses asingle frequency cutoff in the transformed chemical potentialfields ω(k). The newer method used here subdivides frequencyspace into distinct regions, each with its own frequency cutoffvalue. This new spectral filtering algorithm is implemented in aparallel framework in the PolySwift++ code. For all SCFTsimulations employed in this study the following protocol isused. For each one of the four samples, a series of simulations isrun with different filtering strengths for fcut = f 0.6, 0.4, 0.2, and0.0. Typically, the most well-order structure has the lowest free-energy as well. For samples whose location in phase space isnear a boundary between different space groups, it has beenobserved that the spectral filtering algorithm can bias the sampletoward the incorrect morphology. Ultimately, the free-energy ofthe bulk system for each of the fcut values must be compared andthe run with the lowest free-energy value is chosen as the mostlikely space group that will be obtained in the experiments.In order to make comparisons with experimental data the

energy and length scales must be matched to the SCFT simula-tion parameters. The length scale enters the theory through theunperturbed radius of gyration:

(7)

where N is the chain length (number of independent chainsegments) and b is the statistical segment length. The energy

scale enters through the temperature dependent Flory−Hugginsparameters χ(T), which have been reported for poly(isoprene),poly(styrene), and poly(ethylene oxide)37

(8)

based on a common segment volume of 118 A3. Segregationstrength in the SCF theory is proportional to χN, which hasbeen calculated for each of the six samples under considerationusing the molecular weights and block volume fractions (f)listed in Table 1.

IV. RESULTS AND ANALYSISFigure 2 shows a three-dimensional phase map for SISOtetrablock terpolymers. This depiction does not include the role

of segregation strength, χN, which forms a fourth axis. BecauseSISO tetrablocks are prepared from a common (parent) SIStriblock, each set of materials lies on an isopleth with constantf S and f I and varying f O. Also, there is another importantsymmetry parameter associated with these materials: The ratioof poly(styrene) block molecular weights (proportional to xand x′ in Figure 1). In this (and a previous) report we focus ontetrablocks with x = x′.Figure 2 contains two isopleths, one with f S = 1 − f I = 1/2,

which was considered in an earlier publication,38 and a secondfor the case presented here, f S = 0.39. These polymers weredesigned to bridge the order−disorder transition withincreasing O content based on the trends reported by Eppset al.24 for ISO triblocks prepared from a parent IS diblock withf S = 0.42 and Mn = 12.7 kg/mol and 0 ≤ f O ≤ 0.35. Thus, theparent SIS compound (Table 1) was designed with 1.8 timesthe overall molecular weight of the disordered IS diblock,24

precisely the ratio that theoretically corrects for the architecturaldifference.39 In this section, we present and evaluate

Figure 2. Poly(styrene-b-isoprene-b-styrene-b-ethylene oxide) (SISO)phase portrait in the vicinity of the order−disorder transition, which isindicated by the dashed curve. Filled and open circles indicate orderedand disordered states, respectively, within the experimental temper-ature range 100 ≤ T ≤ 250 °C. Specimens considered in this workwere prepared by adding O blocks to a SIS triblock containing 40% Sand 60% I as identified by the line. The isopleth with 50 vol % S and50 vol % I was published elsewhere.38

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experimental and theoretical results that establish the phasebehavior and structure of this set of SISO materials.Order and Disorder. SAXS results acquired at 120 °C

(110 °C for SIS and SISO-1) are presented in Figure 3. These

scattering patterns fall into two categories. SIS and SISO-1 eachproduced a single broad peak consistent with a state of disorder,while SIS-2, -3, -4, and -5 generated three or more well-defineddiffraction peaks with relative positions (q/q*)2 = 1, 3, 4, 7, 9,12, which we associate with a state of hexagonal order.Dynamic mechanical spectroscopy data support the con-

clusions drawn from Figure 3 regarding order and disorder ashighlighted in Figure 4. SISO-1 displays terminal viscoelastic

behavior (G′ ∼ ω2 and G″ ∼ ω) above 80 °C (the lowestfeasible measurement temperature beyond the polystyrene glasstransition) while the remaining tetrablocks are decidedlynonterminal consistent with a triply periodic ordered

morphology between 80 and 160 °C. These trends areanticipated by the corresponding ISO results;24 i.e., adding anO block to SIS leads to a rapid rise in TODT with increasing Omolecular weight. However, whereas the ISO triblocks wereshown to have either tricontinuous cubic (Q230) or lamellar(LAM3) order, the SISO isopleth contains exclusively hexagonalorder.Representative TEM images obtained from SISO-2, -3, -4,

and -5 are presented in Figure 5. Samples were annealed in a

vacuum oven for at least 1 day (120 °C for SISO-3 and 160 °Cfor SISO-2, SISO-4 and SISO-5) prior to quenching into liquidnitrogen to preserve the melt morphology.These images reveal a surprising result. While SISO-5 (f O =

0.32) contains a cylindrical morphology, as would be expectedbased on the hexagonal symmetry identified by SAXS(Figure 2), this is clearly not the case for the three otherordered materials (0.12 ≤ f O ≤ 0.19). Dozens of TEM imagestaken from numerous thin sections obtained from thesepolymers are all consistent with spherical microdomains insharp contrast with the network structures reported for theanalogous ISO compounds.40 Micrographs taken from SISO-2,-3, and -4 each exhibit regions with distinct 6-fold symmetrydefined by well delineated circular white domains distributedon a continuous black matrix. The picture from sample SISO-3shows two well ordered grains that intersect at a grainboundary. Significantly, the grain at the top of this image istilted such that planes of spheres are rotated away from the6-fold axis resulting in a projection that reveals black and

Figure 3. Synchrotron SAXS data for the series of SISO tetrablockcopolymers with a fixed ratio S:I of 4:6 and O content ranging from0 to 32% by volume. Samples were prepared by annealing at 120 °C for1 day, quenched in liquid nitrogen, reheated, and annealed at 120 °C for5 min before taking the measurements. The arrows identify relative peakpositions associated with hexagonal symmetry. Scattering patterns havebeen progressively shifted vertically for clarity.

Figure 4. Dynamic elastic (G′) and loss (G″) moduli measured at120 °C (110 °C for SISO-1) as a function of frequency. These resultsindicate states of order except for SISO-1, which is disordered.

Figure 5. TEM micrographs obtained from OsO4 stained SISOtetrablock copolymers. Key: (a) SISO-2 (160 °C, 1day), (b) SISO-3(120 °C, 1day), (c) SISO-4 (120 °C for 1 day), and (d) SISO-5 (160 °Cfor 3 days), where the temperature and time in the parentheses indicatethe annealing conditions prior to microtoming. Panels a−c containordered arrays of spherical microdomains while panel d is consistent witha cylindrical morphology. Transition from 6-fold to 4-fold symmetricprojections (circled in panel a) that share common planes of spheres isconsistent with a simple hexagonal lattice. Panel b contains two largegrains separated by an unusual grain boundary structure that is similar tothe sigma phase morphology.10

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modulated white layers. This projection confirms that the crystalis not comprised of cylinders; based on the estimated thicknessof the microtomed sections (ca. 60 nm) and the spherediameter (16 nm) we would expect to observe up to three layersof spherical domains. Thus, we conclude that the hexagonalarrangements (6-fold symmetry) derive from vertically alignedstacks of spheres. Also, the grain boundary structure (whichresembles the recently reported σ-phase packing symmetry, seebelow)10 is not compatible with a cylindrical morphology.Additional insight into the hexagonal packing geometry can

be gleaned from Figure 5a, which contains regions with 6-foldand 4-fold symmetry. The square arrangements are accessedfrom the hexagonal ones along specific planes of spheres, whichidentify the axis of rotation required to transform from oneprojection to the other. We compare two possible packingscenarios corresponding to the space groups P6/mmm (primaryor simple hexagonal) and P63/mmc (hexagonal close packed orHCP) in Figure 6 projected along the c (6-fold) axis and rotated

90° around the normal to the (100) plane. These images havebeen constructed using a 30% volume fraction of spheres (f Oplus f S/2 for SISO-3, see below). Clearly, HCP packing doesnot account for the experimental results. Alternating (ABAB...)hexagonal stacking of spheres leads to a projected image of blackdomains on a white matrix and rotation around the [100]

direction fails to produce a 4-fold projection. Simple (primary)hexagonal packing with the placement of one sphere on eachlattice point accounts for the 6-fold TEM projections and settingc = a√3/2 (see below) generates the square projections asshown in Figure 6.The choice of c/a impacts the allowed Bragg reflections

accessed by the SAXS experiments. Table 2 lists the relativepositions of the first nine allowed (hkl) reflections for P6/mmmwith c = a, c = a√3/2 and c = (1/2)a along with HCP(P63/mmc with c = 1.63a). All four hexagonal systemsanticipate the experimentally documented (hk0) reflectionsevident in Figure 3. However, between one and four additionalpeaks associated with (hkl) scattering (l ≠ 0) are permittedover the available q range. We have definitively disqualifiedHCP on the basis of the TEM results and do not furtherconsider this option. TEM appears to favor the ratio c = a√3/2based on the square projection along the a axis (see Figures 5aand 6). This assignment brings the (001) Bragg powder peakinto coincidence with the (100) and the (111) and (002) intoalignment with the (200).However, two additional allowed reflections are absent,

(101) and (201) at relative positions (q/q*)2 = 2 and 5,respectively. The ratio c/a = 1/2 results in a different set ofcoincident reflections and just one absent scattering peak,(111) at (q/q*)2 = 6. With c = a there are four missing peaks.While the combined SAXS and TEM data establish P6/mmmsymmetry we cannot assign unit cell dimensions with certaintydue to the lack of (hkl) diffraction. On the basis of the regionsof 4-fold symmetry seen in the TEM images in Figure 5a, wetentatively assign the lattice parameter c = a√3/2 to thehexagonally packed sphere phase and make use of this result inthe remainder of this report. Nevertheless, we cannot rule outother possibilities without additional diffraction data.One explanation for the absence of (hkl, l ≠ 0) diffraction

may be a lack of long-range order (i.e., weak correlations) alongthe 6-fold axis (i.e., the [001] direction) relative to the strongin-plane (hk0) order. A complete loss of correlation betweenstacks (lines) of spheres would result in a columnar structureand eliminate (hkl, l ≠ 0) scattering. Obviously, there is somedegree of correlation as clearly evidenced by Figure 5a, whererotation around the (100) plane leads to patches of 4-foldsymmetry. Nevertheless, the 6-fold coordination is generallymuch more prominent than the 4-fold packing in these images,consistent with the argument of weaker stacking correlations.Block Segregation. A tetrablock molecular architecture

affords additional levels of microphase complexity over thoseavailable to triblock terpolymers, including segregation of the O,terminal S and interior S′ blocks from I-rich domains. The actualstate of microdomain segregation cannot be simply determined byTEM or SAXS without making assumptions regarding the unit celldimensions (see previous paragraphs). DSC results shown inFigure 7 reveal endotherms between 20 and 70 °C that can beattributed to the melting of O crystals during heating for all fivetetrablocks, with calculated crystallinities ranging from 36 to 71%(Table 1).These results indicate that the thermodynamically incompat-

ible O blocks microphase separate in the melt state. (SISO-1 isan exception since the melting temperature Tm,O deviates fromthe trend set by the other specimens, i.e., a decreasing Tm,O withO molecular weight. We attribute this behavior to crystal-lization from the homogeneous disordered state.) Segregationof the S blocks is difficult to establish by DSC due to thepresence of the melting endotherms. Subtle evidence of an

Figure 6. Simulated projections of hexagonal crystal structures formedfrom spherical (30% by volume) domains: P6/mmm with c = a√3/2(left) and P63/mmc (right). The top images represent projectionsalong the 6-fold (c) axis, while the bottom pictures were obtained byprojecting along the a axis, i.e., after rotating the crystal by 90°.Comparison of these images with the TEM results in Figure 5indicates that the spheres pack on a simple hexagonal lattice.

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S block glass transition is evident in the thermal trace for thedisordered SIS triblock (inset to Figure 7) suggestive offluctuation effects41 or possibly microphase separation below80 °C. Every specimen displays a glass transition at about−60 °C attributable to I-rich domains.Three distinct limiting states of melt segregation can be

postulated for the SISO system as illustrated in Figure 8: (a) asegregated O core immersed in a matrix containing ahomogeneous mixture of S, S′ and I; (b) an O core surroundedby a segregated S′ shell distributed in a mixed S and I matrix;(c) an O core surrounded by a shell of segregated S and S′ witha continuous matrix of I. Critical insights into the state ofsegregation and the transition from a spherical to cylindricalmorphology are provided by the SCFT calculations.Figures 9 and 10 summarize results obtained at T = 120 °C

for f O = 0.12 (corresponding to SISO-3) and fO = 0.32(corresponding to SISO-5). The theory correctly anticipatesthe transition from a spherical (BCC lattice) to cylindrical(hexagonal lattice) morphology with increasing O content.Here we focus on the distribution of block segments acrossa unit cell, which is difficult to characterize experimentally.Although the theory does not account for the experimentallydocumented spherical packing symmetry, the composition

profile around each domain is relatively insensitive to the exactstate of order.Figure 9 illustrates variations in the volume fraction of

poly(styrene) segments computed along two directions thateach intersect the sphere centers: [111] (direction of closestpacking in the BCC lattice, Figure 9a) and [100] (Figure 9b).The theory allows us to separate the location of the inner andouter S segments (S′ and S, respectively) as a function ofposition. These calculations reveal that the predicted placementof poly(styrene) segments represents a hybrid of models (2)and (3) shown in the Figure 8. Most of the inner S′ segmentsare located in a shell around a mostly segregated O core whilethe outer S segments are distributed between the S shell andthe I rich matrix. Comparison of the computed compositionprofiles at different locations in the unit cell demonstrates thatthe S′ repeat units are dispersed rather uniformly in the radialdirection away from the spherical cores; i.e., the concentrationmidway between the sphere centers at 1/2,

1/2,0 position (along[100]) and at the 1/4,

1/4,1/4 (along the line of closest packing

[111]) are nearly equivalent. We believe that the constraintsimposed by optimal placement of the inner S′ and outer Sblocks are responsible for the experimentally determinedhexagonal lattice as discussed in the following section.We have estimated the area fraction of white domains in

parts a−c of Figure 5 using the best ordered (hexagonal)sections of the micrographs. These have been converted tovolume fractions assuming c = a√3/2 and the results are listedin Table 3 along with the values expected for models 1−3(Figure 8). On the basis of this analysis the volume fraction ofwhite spherical domains ranges from 20 to 30%, somewhat lessthan the sum of the O and S block volume fractions. Comparisonof these measurements with the theoretical predictions iscomplicated by the staining process used to obtain TEMcontrast. The osmium tetraoxide reacts selectively with poly-(isoprene) creating the dark regions. As shown by the theoreticalcalculations, we expect a sharp but continuous gradient of

Table 2. Allowed Reflections for Different a and c Relationships

(100) (001) (110) (101) (111) (200) (002) (201) (210)

expt data √1 √1 √3 - √4 √4 √4 - √7c = a √1 √1.5 √3 √3.5 √7.5 √4 √3 √9.5 √7c = a√3/2 √1 √1 √3 √2 √4 √4 √4 √5 √7c = a/2 √1 √3 √3 √4 √6 √4 √3 √7 √7c = 1.63a (HCP) √1 - √3 √1.28 - √4 √1.13 √4.28 √7

Figure 7. DSC traces obtained from the SIS triblock and SISOtetrablocks. Curves have been shifted vertically for clarity. Endothermsbetween 20 and 70 °C reflect melting of poly(ethylene oxide)consistent with segregation of the O blocks in a core−shellmorphology. The small peak at 55 °C for SISO-1 is attributed tocrystals formed from the homogeneous melt. All the specimens displaya glass transition at −61 °C, which is associated with poly(isoprene).The inset shows evidence of a poly(styrene) glass transition in SIS,suggestive of composition fluctuations or segregation at T < 65 °C forthis specimen.

Figure 8. Discrete states of microphase separation for sphere formingSISO tetrablock copolymers. (1) Segregation of O blocks from mixedinner and outer S and I blocks. (2) Segregated core of O surroundedby a shell of inner S blocks embedded in a matrix of mixed I and outerS blocks. (3) Core of O surrounded by a shell of segregated inner andouter S blocks embedded in a matrix of I. Self-consistent field theorycalculations indicate that the actual morphology is a hybrid of parts2 and 3 as shown in Figure 9.

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I blocks from the matrix toward to spherical cores, terminatingjust beyond the O−S interface. Assuming the stain partiallypenetrates the S and S′ rich shell and reacts with I in theinterfacial region, the estimated core volume fractions areroughly consistent with the theoretical values.Figure 10 shows a representative SCFT calculation for the

fO = 0.32 case, which yielded a hexagonally packed cylindricalmorphology formed by an O core, a shell of S and S′, and an Irich matrix. As with the spherical geometry, the inner S′ andouter S blocks are distributed asymmetrically around the Ocores, with a significant fraction of the S blocks mixed into the Imatrix.

V. DISCUSSION

The results presented in the previous section expose aremarkable finding: Ordering of spherical microdomains on asimple hexagonal (P6/mmm space group) lattice. We have

become accustomed to BCC packing of single componentasymmetric block copolymers as anticipated by Leibler42 morethan 30 years ago based on self-consistent field theory andconfirmed experimentally soon thereafter.43 This behavior canbe rationalized on the basis of minimizing the extent of chainstretching and compression driven by the constraint ofmaintaining a uniform density due to the incompressibility ofpolymer melts. Simple geometric calculations show that closelypacked spheres (FCC and HCP) fill space with a more hetero-geneous distribution of matrix space (i.e., wider range ofrelative spacing between spheres) than a BCC lattice, hence thelater is favored with single component block copolymer melts.Simple hexagonal ordering is rare across all areas of materials

science and engineering, as exemplified by the elements in theperiodic table. Most elements pack onto three types of lattices:FCC, HCP, and BCC. (Covalently bonded compounds such asCl2, O2, and sulfur tend to crystallize with orthorhombicsymmetry, while carbon forms either diamond or graphite.)There are a few exceptions to this trend including the group VBelements As, Sb and Bi, along with Hg, which solidify intorhombohedral crystals, and tetragonal In and Sn. Crystalsymmetry is controlled by the detailed nature of the interatomicpotential, which depends in a complex way on the electronic

Figure 9. SCFT simulated density profiles of O, terminal S andinterior S′ blocks against position variable, r, for a model of the SISO-3tetrablock copolymer. The calculation yields BCC symmetry, contraryto the experimental result, which is attributed to the mean-field natureof the theory. Top panel: [111] unit cell direction denoted by the redline in the inset. Bottom panel: [100] unit cell direction denoted bythe blue line in the inset. These results indicate that the SISOtetrablock copolymers segregate with a distribution of blocks that is ahybrid of models 2 and 3 illustrated in Figure 8.

Figure 10. SCFT simulated density profiles of O, terminal S andinterior S′ blocks against position variable, r, for a model of the SISO-5tetrablock copolymer with hexagonal (P6/mm) symmetry. Densityprofiles were calculated along the [100] unit cell direction as shown inthe inset.

Table 3. Comparison between Calculated and ExperimentalSpherical Volume Ratio

% theoretical volumea experimental resulte

sample PEOPEO+1/2

PSPEO+PS a (nm)b,c d (nm)b,c

volume(%)d

SISO-2 9 27 44 20.0 ± 0.8 13.3 ± 1.0 20 ± 5SISO-3 12 29 46 22.6 ± 1.2 16.0 ± 1.2 25 ± 7SISO-4 19 35 51 27.2 ± 0.7 20.6 ± 1.2 30 ± 6aTheoretical spherical volume fractions based on experimental datafrom Table 1. bLattice dimension and sphere diameter are obtainedfrom TEM images for SISO-2, -3, and -4 respectively. c could becalculated using c = a√3/2. cTemperature is 120 °C. dVolumefraction is the volume ratio between the sphere and the hexagonal unitcell. eOn the basis of the assumption that c = a√3/2. The indicateduncertainty does not include the possibility that c ≠ a√3/2.

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configuration and the underlying orbital symmetries. Subtleeffects in certain heavy elements produce the most exotic results.For example, simple cubic packing in polonium has beenattributed to relativistic corrections to the effective mass ofvalence electrons,44 while uranium and tantalum form a σ-phasewith 30 atoms per tetragonal unit cell.45 We are not aware ofany reports that assign the P6/mmm space group to singlecomponent crystalline materials, although certain intermetalliccompounds (e.g., CaCu5) order with this symmetry.46

Ordered sphere forming block copolymers appear to offer afascinating counterpart to monatomic crystals. We believe theinterparticle potential that controls sphere packing is sensitiveto the overall corona block molecular weight and architecture.In a recent publication, we have shown that a low molecularweight poly(isoprene-b-lactide) (IL) diblock and a SISOtetrablock copolymer can form the σ-phase, a crystalapproximant to dodecagonal quasicrystals.10 The results reportedhere represent another variation on this theme, driven bychanges in the composition of the multiblock copolymer. Thesequence of blocks in SISO enforces a spherical morphologyover a wider range of sphere volume fractions (relative todiblocks) as a consequence of the unfavorable segment−segmentinteractions between I and O blocks. Formation of a shell rich inS (and S′) segments screens these contacts. However, the matrixmaterial contains S, I, and S′ segments distributed anisotropicallyaround the core and shell structure as suggested by the SCFTcalculations. Unfortunately, the field theory calculations are notreadily transformed into an effective interdomain interactionpotential that might be employed to model ordering. Never-theless, we can draw a loose yet provocative analogy betweenordering in sphere forming block copolymers, which is governedby statistical mechanical concepts, and crystallization of theelements, which requires incorporation of sophisticated quantummechanical principles. (Here we note that the number of blocksper sphere can vary somewhat (as shown for the σ-phase10)while the atomic structure of each element is invariant. On theother hand, electrical charge generally is not distributeduniformly throughout the unit cell of an elemental crystal.)Changing the composition and molecular weight of the corona

blocks, which affects the segregation strength, will influence thetendency to order on a particular lattice. To date there are atleast three documented examples: Im3m̅ (BCC), P42/mnm(σ-phase), and P6/mmm. Fundamental arguments supporting thegeneral occurrence of BCC and icosahedral packing (closelyrelated to the σ-phase) were advanced by Alexander andMcTague47 based on Landau theory and subsequently applied todiblock copolymers by Roan and Shakhnovich using the density-wave mean field theory developed by Leibler.48,49

There are a variety of reasons why the present SCFTsimulations did not precisely reproduce the observed hexagonalspherical packing morphology. Fluctuation effects are notincluded in the current mean-field formulation of the SCFTsimulations. For dense copolymer melts fluctuation effects aremost relevant near phase boundaries between morphologies.Fluctuation effects are known to shift the location of phaseboundaries, but have not heretofore been shown to createordered phases with new space groups. Our conjecture for thesource of this discrepancy between experiment and simulationis the Gaussian chain model used in the present SCFTalgorithms. Including finite-chain length effects could changethe effective domain−domain interactions by introducinganisotropic forces between these spherical domains. This isrelated to the analogy made above to the novel crystalline order

seen in some metals with large atomic number and anisotropicouter shell electronic orbitals. Extending the model to includefinite chain length effects, and improving the numericalmethods to find these novel phases, will be the focus of futurework.

VI. CONCLUSIONWe have demonstrated in this work that a SISO tetrablockterpolymer molecular architecture introduces levels ofmorphological control unavailable to more conventional ISOtriblocks. Combined use of synchrotron SAXS and TEMconclusively demonstrate P6/mmm space group symmetry forspecimens containing a fixed ratio, f S/f I = 2/3, of poly-(isoprene) (I) and poly(styrene) block volume fractions (equallength S blocks) and between 9 and 32% poly(ethylene oxide)(O). Unfavorable segment−segment interactions between the Iand O blocks drive the formation of spherical microdomains forf O = 0.09, 0.12, and 0.19, while a fO = 0.32 specimen contains acylindrical morphology. Self-consistent mean-field theorycalculations correctly anticipates the core (O) and shell (S)domain geometries but fail to account for simple hexagonalpacking of the spheres. Nevertheless, these calculations provideimportant insight into the distribution of block segments withinthe ordered structures including the asymmetric placement ofinterior and terminal S blocks within the core, shell and matrixdomains. These results, together with an earlier publication,38

demonstrate a powerful approach to manipulating blockcopolymer structure outside the traditional paradigms (AB,ABA, ABC) with respect to composition.

■ ACKNOWLEDGMENTSThis work was supported by the U.S. Department of Energy,Basic Energy Sciences, Division of Materials Science andEngineering, under Contract Number DEAC05-00OR22725with UT-Battelle LLC at Oak Ridge National Laboratory. Someof the facilities employed derived support from the Universityof Minnesota Materials Research Science and EngineeringCenter (MRSEC) (NSF-DMR-0819885). Portions of this workwere performed at the DuPont−Northwestern−Dow Collab-orative Access Team (DND-CAT) located at Sector 5 of theAdvanced Photon Source (APS). DND-CAT is supported byE. I. du Pont de Nemours & Co., the Dow Chemical Company,and the State of Illinois. Use of the APS was supported by theU.S. Department of Energy, Office of Science, Basic EnergySciences, under Contract No. DE-AC02-06CH11357. Parts ofthis work were carried out in the University of MinnesotaCollege of Science and Engineering Characterization Facility,which receives partial support from NSF through the NNINprogram. We thank Fang Zhou for her assistance inmicrotoming the TEM sections.

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■ NOTE ADDED AFTER ASAP PUBLICATIONThis article posted ASAP on December 7, 2011. Equation 2 hasbeen revised. The correct version posted on December 8, 2011.

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