Origins of low-symmetry phases in asymmetric diblockcopolymer meltsKyungtae Kima, Akash Aroraa, Ronald M. Lewis IIIa, Meijiao Liub, Weihua Lic, An-Chang Shid, Kevin D. Dorfmana,and Frank S. Batesa,1

aDepartment of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455; bDepartment of Chemistry, Key Laboratoryof Advanced Textile Materials and Manufacturing Technology of Education Ministry, Zhejiang Sci-Tech University, Hangzhou 310018, China; cState KeyLaboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China; and dDepartment ofPhysics and Astronomy, McMaster University, Hamilton, ON L8S 4M1, Canada

This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected in 2017.

Contributed by Frank S. Bates, December 21, 2017 (sent for review October 11, 2017; reviewed by Sol M. Gruner and Zhen-Gang Wang)

Cooling disordered compositionally asymmetric diblock copolymersleads to the formation of nearly spherical particles, each containinghundreds of molecules, which crystallize upon cooling below theorder–disorder transition temperature (TODT). Self-consistent fieldtheory (SCFT) reveals that dispersity in the block degrees of poly-merization stabilizes various Frank–Kasper phases, including theC14 and C15 Laves phases, which have been accessed experimen-tally in low-molar-mass poly(isoprene)-b-poly(lactide) (PI-PLA)diblock copolymers using thermal processing strategies. Heatingand cooling a specimen containing 15% PLA above and below theTODT from the body-centered cubic (BCC) or C14 states regeneratesthe same crystalline order established at lower temperatures. Thismemory effect is also demonstrated with a specimen containing20% PLA, which recrystallizes to either C15 or hexagonally or-dered cylinders (HEXC) upon heating and cooling. The process-path–dependent formation of crystalline order shapes the numberof particles per unit volume, n/V, which is retained in the highlystructured disordered liquid as revealed by small-angle X-ray scat-tering (SAXS) experiments. We hypothesize that symmetry break-ing during crystallization is governed by the particle numberdensity imprinted in the liquid during ordering at lower tempera-ture, and this metastable liquid is kinetically constrained fromequilibrating due to prohibitively large free energy barriers formicelle fusion and fission. Ordering at fixed n/V is enabled byfacile chain exchange, which redistributes mass as required tomeet the multiple particle sizes and packing associated with spe-cific low-symmetry Frank–Kasper phases. This discovery exposesuniversal concepts related to order and disorder in self-assembledsoft materials.

structured liquids | fluctuating disorder | Frank–Kasper phases |block copolymers | sphere-forming diblocks

“There are several ways,” Dr. Breed said to me, “in which certainliquids can crystallize—can freeze—several ways in which their atomscan stack and lock in an orderly, rigid way.”

Chapter 20, Cat’s Cradle by Kurt Vonnegut (1)

Liquids are a fascinating state of matter, with densities close tothose of the corresponding perfectly ordered crystals but macro-

scopically disordered like a gas. When cooled, a liquid freezesthrough nucleation and growth of periodic structures that reduce thefree energy of the system relative to the isotropic state. Symmetrybreaking during crystallization affords many options for tiling spaceresulting in an infinite array of possible periodic and aperiodicstructures. Even simple monoatomic metals can assume multipleordered phases as a function of temperature and pressure. Iron formsa body-centered cubic (BCC) phase just below the melting temper-ature and then transforms into a close-packed face-centered cubic(FCC) state and back to BCC when cooled to room temperature (2).

Manganese transitions through four different crystalline phasesupon cooling from the melt, forming a complex low-symmetrycrystal with a unit cell containing 58 atoms in the ground state(3–5). Plutonium exhibits six different crystalline phases atambient pressure (and one more at elevated pressures); re-markably, near the melting temperature the solids float on theliquid (6). Mixtures of metals, referred to as alloys, can be evenmore complex (7), generating aperiodic crystals known asquasicrystals with certain formulations (8, 9). Molecules intro-duce additional complexity. Water, a deceptively simple triatomicmolecule, is known to exhibit 18 different crystalline phases (10,11). Understanding how larger compounds (e.g., liquid crystals,pharmaceuticals, or proteins) adopt ordered states presentsdaunting challenges. Moreover, large crystals are easily trappedin metastable states due to kinetic barriers associated with thenucleation and growth of equilibrium phases in the solidstate (12).Liquids are different. Lacking long-range order, and endowed

with fluid properties, dense single-component liquids are gen-erally characterized by a dynamically accessible minimum freeenergy (except near the gas–liquid critical point) with respect tovariables such as density or density distribution, resulting in asingle spatially uniform state at fixed temperature and pressure.This report describes experimental results that challenge theuniversality of this concept. We demonstrate that self-assembled

Significance

We demonstrate that low-molecular weight asymmetric diblockcopolymer melts can form multiple metastable liquid states at acommon temperature, dependent on the processing history. For-mation of ordered self-assembled micelles at low temperaturesshapes the number density of the mesoscopic particles, whichis preserved upon heating above the order–disorder transitiontemperature. Cooling returns the liquid to the same crystallinestate reflecting amemory—a type of hidden symmetry—imprintedin the fluid. These surprising results are explained based on thelarge energetic penalty associated with fusing or fragmentingmicelles in the highly structured liquid state. This work revealsconcepts related to spontaneous symmetry breaking in self-assembled soft materials including surfactant-based systems.

Author contributions: K.K., A.A., R.M.L., M.L., W.L., A.-C.S., K.D.D., and F.S.B. designedresearch; K.K., A.A., R.M.L., M.L., W.L., A.-C.S., and K.D.D. performed research; K.D.D. andF.S.B. supervised research; K.K., A.A., R.M.L., K.D.D., and F.S.B. analyzed data; and K.K.,A.A., R.M.L., K.D.D., and F.S.B. wrote the paper.

Reviewers: S.M.G., Cornell University; and Z.-G.W., California Institute of Technology.

The authors declare no conflict of interest.

Published under the PNAS license.1To whom correspondence should be addressed. Email: bates001@umn.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1717850115/-/DCSupplemental.

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disordered diblock copolymer liquids produce multiple metastablenonequilibrium ordered states upon cooling whose symmetrieswould not arise if the liquid had a single spatially uniform state.Diblock copolymers are perhaps the simplest, and experimen-

tally and theoretically most tractable, amphiphilic compounds.This class of materials includes lipids (13–15), surfactants (16–19),dendrimers (20–22), and various other types of hybrid molecules(23–26). (Here we broaden the traditional definition of an am-phiphile to embrace compounds beyond those that exhibit partialcompatibility with water.) Self-assembly into nanoscale objectsresults in mesoscopic structures that can be arranged into aplethora of morphologies, subject to the constraint that space isfilled at uniform density. In general, soft materials—such as theflexible nonionic A–B diblock copolymers considered here—areheld together by short-range forces, predominately van der Waalsinteractions. Preferential self-association of chemically different Aand B blocks with degrees of polymerization NA and NB leads todomains with dimensions that scale as Nδ, where 1/2 ≤ δ ≤1 depending on the segregation strength and chain flexibility, andN = NA + NB (27). The geometry of the domains is controlledprimarily by the composition (volume fraction) of each polymerblock, fA = NA/N, and by conformational asymmetry, e = (bA/bB)

2,

where bj = Rg,j(6/Nj)1/2 (based on unperturbed Gaussian chain

statistics for flexible polymers), defined using a common repeatunit (segment) volume v0 (28). We refer the reader to an extensiveliterature, developed over the last half century, which describeslamellar, multiply continuous network, cylindrical, and sphericalordered phases in diblock copolymer melts (27–31).The work reported here focuses on the limit of compositional

asymmetry, fA << 1/2, where discrete particles form at suffi-ciently high values of χN in which χ ∼ T−1 represents the Flory–Huggins segment–segment interaction parameter. Until recently,it was widely accepted that undiluted flexible diblock copolymersbehave in a simple and universal manner, forming a BCC ar-rangement of spherical particles at equilibrium below the order–disorder transition temperature (TODT) and a disordered statefor T > TODT (32). Discovery of the Frank–Kasper σ phase in2010 (33) and subsequent reports of a dodecagonal quasicrystal(DDQC) (34) and the C14 and C15 Laves phases (35) in shortasymmetric poly(isoprene)-b-poly(lactide) (PI-PLA) diblock co-polymers, along with several theoretical advances (36, 37), havedisrupted this understanding.Fig. 1 illustrates the structure of asymmetric diblock copoly-

mers as a function of temperature above and below TODT. Self-consistent field theory (SCFT) predicts that the difference infree-energy per chain between various Frank–Kasper phases isextremely small, O(10−3 kBT), and that such tetrahedrally close-packed structures are favored over BCC order for certain com-binations of fA, e, and χN (35). Modest dispersity in the blockmolar masses, which is unavoidable in most experimental mate-rials, also has been shown to influence the predicted equilibriumphase behavior, for example, favoring formation of the σ phaserelative to BCC (37). In this report, we demonstrate using SCFTthat certain levels of dispersity can account for all of the orderedphases shown in Fig. 1 at equilibrium.At T > TODT, SCFT anticipates a disordered phase charac-

terized by a uniform, homogeneous composition over lengthscales d >> b (32). This mean field treatment does not accountfor the effects of composition fluctuations, which dominate thedisordered liquid state structure near TODT at finite molarmasses (Fig. 1). At sufficiently high temperatures the disorderedliquid will be spatially uniform in composition as entropic factorsoverwhelm unfavorable enthalpic contributions (χ > 0) to theoverall free energy. As the temperature is reduced, a point isreached where micelles begin to form, referred to as the micelledissociation temperature (TMDT) (38–40). Reducing the tempera-ture further increases the concentration of micelles, which even-tually crowd and fill space. The size of the fluctuating disordered

region, ΔT = TMDT − TODT, and the amplitude of the compo-sition fluctuations scale with molar mass (28, 40): Δ(χN) =(χN)MDT − (χN)ODT ∼ �N−γ, where �N = N(b3/v0)

2 and 1/3 ≤ γ ≤1/2; the theoretical value for γ when fA << 1/2 has not beenfirmly established (40). In the mean-field limit (i.e., �N → ∞),ΔT → 0. Increasing χ by choosing thermodynamically moreincompatible blocks mandates reducing �N to achieve an experi-mentally viable TODT, which drives up ΔT. For short asymmetricPI-PLA diblock copolymers the disordered liquid is highlystructured, displaying segregated micelles over a sizable range oftemperatures, ΔT > 15 °C, as evidenced by rheological andsmall-angle X-ray scattering (SAXS) measurements (5).Recently, we discovered that the ordered-state symmetry that

evolves from a disordered PI-PLA liquid is process path dependent(35). Rapidly cooling the disordered liquid by immersion in liquidnitrogen and subsequent heating to T* < TODT produced a differentset of ordered phases (including C14 and C15) than were obtainedupon slowly cooling from TODT to T* [BCC, σ, and hexagonallyordered cylinders (HEXC)] (Fig. 1). Based on conventional wisdomwe previously presumed that the disordered liquid rapidly equili-brates regardless of thermal treatment. We have now found that

T

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HEXCBCC

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Fig. 1. Schematic of ordered and disordered regimes in diblock copolymermelts in the idealized equilibrium limit. Colored spheres denote particlecores formed by minority blocks, and the blue background indicates thecorona matrix consisting of the majority blocks. Above the micelle dissoci-ation temperature (TMDT) the disordered polymer is spatially homogeneousas accounted for by self-consistent field theory (SCFT), whereas locally seg-regated micelles dominate the disordered state structure near the order–disorder transition temperature (TODT). SCFT anticipates the experimentallyobserved phase behavior illustrated at T < TODT.

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cooling the disordered liquid back below TODT returns the materialto the same crystalline state from which it came. In other words,the disordered diblock copolymer has structural memory on thetimescales of the process. Clearly, these fluctuating liquids, whichare nearly pourable fluids, are not at equilibrium. We presentevidence here, drawn from SAXS experiments that show that theaverage particle (micelle) size in the disordered state near TODT iscontrolled by the state of aggregation imprinted on the materialduring ordering. Moreover, micelle density appears to be a conservedquantity above and below TODT on the timescale of the experiment,dictating the ordered phases accessed upon cooling. We argue thatthis seemingly perplexing behavior is a natural consequence of thesevere constraints placed on particle fusion and fission due to theenergetic penalties associated with merging two micelles ordividing a micelle into smaller pieces.

Results and AnalysisBeginning with Infinity: Self-Consistent Field Theory. SCFT is anideal starting point for understanding the equilibrium phasebehavior of diblock copolymers. As a mean-field theory, SCFT isstrictly valid in the limit of infinitely large chains ( �N → ∞) (41).However, because fluctuation effects due to finite molar massonly manifest near the order–disorder transition, SCFT providesreasonable accuracy for determining the free energy differencebetween ordered phases, which is our interest here.For compositionally asymmetric but conformationally sym-

metric diblock polymers, SCFT calculations dating back to Leibler’sseminal work (32) predict that BCC is the stable phase for nomi-nally sphere-forming compositions, with a narrow window of FCCnear the order–disorder transition (where SCFT is not as reli-able). Following the 2010 discovery of the σ phase (33), Shi andcoworkers (36, 37) used SCFT to uncover two mechanisms thatstabilize the σ phase relative to BCC. First, conformationalasymmetry opens up a σ-phase window at relatively high valuesof χN (36). The compositional asymmetry required to achieve astable σ phase in SCFT, e = 2.25, is large. Nevertheless, bypreparing diblock copolymers with conformational asymmetries ofe = 1.06, e = 1.32, and e = 1.68, Schulze et al. (42) demonstratedexperimentally that the σ-phase window indeed expands as con-formational asymmetry increases, with no σ phase appearing for themost conformationally symmetric system. Second, blending ofdiblock copolymers with disparate compositions (Fig. 2A) shouldalso favor the formation of the σ phase relative to BCC, as wellas a region of stable A15 phase (37). These blending calculationscould explain why experiments found a stable σ-phase region atrelatively low conformational asymmetries compared with SCFT;even a small dispersity in chain lengths and block compositionsmay produce effects akin to binary blending. For both confor-mational asymmetry and blending, Shi and coworkers (36, 37)argue that the additional degrees of freedom in these systemsrelieve the packing frustration required to form the five differentsized particles in the σ phase.One of the challenges in SCFT calculations is identifying a

suitable set of candidate phases in advance (41, 43). Such can-didate phases are suggested often by experimental observations.Thus, having observed the C14 and C15 phases in diblock co-polymers via thermal processing, Kim et al. (35) built upon priorwork (36) by performing SCFT calculations for all experimen-tally known sphere-forming phases in conformationally asym-metric diblock polymers (BCC, FCC, σ, C14, and C15) as well astwo other Frank–Kasper phases (A15 and Z) over a wide rangeof conformational and compositional asymmetries. Althoughmany of the Frank–Kasper phases indeed are stabilized relativeto BCC at some point within the phase space, SCFT only predictsσ as an equilibrium Frank–Kasper phase (35).To determine whether blending is sufficient to stabilize C14 or

C15, we performed here extensive SCFT calculations for the

binary blend of diblock copolymers, A1B and A2B, shown sche-matically in Fig. 2A (37), now including C14 and C15 as candidatephases. Both diblocks are conformationally symmetric and haveequal length of the majority (B) block, whereas the length ofminority block differs. In the mixture, the volume fraction of A1Bis ϕ1, and its block composition was chosen such that it forms aBCC phase in a single-component system. Furthermore, the mo-lecular parameters of A1B are kept constant, whereas the lengthof the A2 block in A2B is varied; increasing N2 implies making theA2 block longer. HEXC is the equilibrium morphology for pureA2B over the range of N2 values considered here.Fig. 2B shows the phase behavior of the blended system in the

N2/N1–ϕ1 plane with the other parameters fixed (see SCFT Calculations

Fig. 2. SCFT calculations for a binary blend of monodisperse AB diblockcopolymers. (A) Schematic of the system formed by blending two diblocks,A1B and A2B, where ϕ1 is the volume fraction of A1B. (B) Phase portraitdepicting the regions of stability of different ordered phases in the N2/N1–ϕ1

plane. (C) Per-chain free energies of the different phases relative to that ofthe BCC phase along the dashed line shown in B (i.e., for N2/N1 = 1.7). Thecalculations were performed with χN1 = 40, NA1 = 0.15N1, and NB = 0.85N1.

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for details). C14 and C15 occur as equilibrium phases within themean-field limit and are stabilized over a wide region of the phasespace. We have only considered conformationally symmetric (e = 1)systems in Fig. 2B, but further calculations suggest that conforma-tionally asymmetric diblocks (e > 1) will stabilize C14 and C15 atmore modest blend compositions than in Fig. 2B (Fig. S1). We alsocomputed the relative stability of the different phases at a fixedblend composition in Fig. 2C. Consistent with previous results forsingle-component systems (35), the free energies of differentFrank–Kasper phases differ marginally, i.e., by only the order of10−3 kBT per chain.

Experiments with Finite-Length Diblock Copolymers. We have ex-tended our recently reported work with PI-PLA diblock copol-ymers, which revealed the formation of the C14 and C15 phases,to include heating and cooling these materials above and belowTODT. Samples IL-58-15 (number-average molar mass Mn =5,010 g/mol, lactide volume fraction fL = 0.15, dispersity Ð =1.08) and IL-52-20 (Mn = 4,830 g/mol, fL = 0.20, Ð = 1.07) werefreeze-dried, hermetically sealed in aluminum DSC pans, andthermally treated as described earlier (35). Directly cooling thedisordered melt to temperatures just below TODT producedstates of BCC (IL-58-15) and HEXC (IL-52-20) order as con-firmed by SAXS patterns nearly identical to those shown in Figs.3A and 4A, respectively. Immersing the disordered specimens inliquid nitrogen followed by reheating to approximately the sametemperatures accessed during direct cooling led to the formationof the C14 (IL-58-15) and C15 (IL-52-20) phases, also confirmedby SAXS data that are virtually indistinguishable from the scat-tering patterns presented in Figs. 3B and 4B, respectively. Theseresults mirror what we reported previously (35). Then we heatedeach DSC pan above the TODT to melt the ordered structuresand cooled them back directly (at about 2 °C/s) to T < TODT.These experiments produced two sets of surprising results.Heating the four ordered specimens to T > TODT generated

SAXS patterns characterized by a single broad peak (Figs. 3 and 4)consistent with a disordered state. However, a striking feature ofeach pair of scattering patterns is that they are characterized bysignificantly different peak positions, q*, where q = 4πλ−1sin(θ/2)

is the magnitude of the scattering wave vector and θ is the anglebetween the incident and scattered X-ray beams. Heating theBCC version of sample IL-58-15 leads to q* = 0.0633 Å−1 at 95 °C,whereas the C14 morphology generates a peak at q* = 0.0538 Å−1

upon heating to 110 °C. This 15% variation far exceeds the smallchange in q* (ca. 1–2%) associated with the difference in tem-perature (Fig. S2). Similarly, heating sample IL-52-20 from theHEXC state to 140 °C produces q* = 0.0657 Å−1, whereas theC15 ordered material when melted generates q* = 0.0506 Å−1 at140 °C, a 23% difference. These sizable variations in the disor-dered state peak positions signal corresponding differences in theaverage particle density in the disordered state. Here we note thatthese disordered diblock copolymers are relatively low-viscosity(<10 Pa·s) liquids (5). Moreover, these differences survive heatingto as much as 30 °C above TODT for 5 min (Fig. S3).Remarkably, all four disordered fluids retain a memory of the

ordered state from which they came. Cooling the disorderedversions of sample IL-58-15 that began as BCC and C14 returnseach specimen to the original state as shown in Fig. 3. This be-havior can be repeated multiple times by heating and coolingthrough TODT (Figs. S3–S5). The BCC and C14 versions of sampleIL-58-15 are characterized by different TODT values as determinedby SAXS: TODT,BCC = 95–99 °C and TODT,C14 = 105–110 °C.Similar behavior was obtained with sample IL-52-20 (Fig. 4).

Following disordering, the HEXC and C15 structures returnupon slowly cooling as shown in Fig. 4. However, recrystallization ofC15 does not appear to be as perfect as with C14. SAXS patternscorresponding to a C15 structure obtained upon slow cooling andannealing appear to have a noticeable content of stacking faults(Fig. S6). Nevertheless, the disordered liquid recovers thegeneral structure formed by quenching the freeze-dried ma-terial in liquid nitrogen followed by reheating. In this caseboth ordered morphologies, HEXC and C15, have a similarTODT of 135–140 °C. We also note that the microdomainsassociated with the HEXC symmetry are assumed to be cy-lindrical (Fig. 1) based on established diblock copolymerphase behavior (44–46). The apparent TODT values obtainedfrom the SAXS experiments for both the IL-58-15 and IL-52-20 samples are summarized in Table S1.

A B

Fig. 3. Small-angle X-ray scattering (SAXS) results obtained from IL-58-15 (Mn = 5,010 g/mol, fL = 0.15, Ð = 1.08). (A) Specimen was freeze-dried and heatedto 95 °C, followed by cooling to 85 °C. Subsequent heating and cooling to these temperatures produced scattering patterns consistent with states of disorderand BCC symmetry. (B) A freeze-dried specimen was heated to 95 °C then immersed in liquid nitrogen and reheated to 95 °C. Heating to 110 °C and cooling to95 °C produced SAXS patterns that correspond to a disordered fluid and the Laves C14 phase, respectively. Data are shifted vertically for clarity. These SAXSresults were duplicated during multiple heating and cooling cycles. The slight hump in the disordered states at q > q* can be coming from the liquid-likepacking arrangement of the disordered micelles (i.e., the structure factor) or from the form factor of the micelles themselves. A direct comparison of thedisordered state scattering patterns with that obtained immediately after heating the freeze-dried material is presented in Fig. S7. Vertical marks identify thepredicted peak positions for the ordered states (35).

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The exquisite diffraction patterns shown in Figs. 3 and 4provide a direct method for calculating the number of particlesper unit volume n/V associated with each ordered phase (TableS2). A BCC crystal (Im�3m space group symmetry) contains twoequivalent particles per unit cell, which corresponds to (n/V)BCC =7.3 × 10−4 nm−3 at 90 °C. The C14 Laves phase (P63/mmc) con-tains three different particle types distributed across 12 sites perunit cell, where (n/V)C14 = 4.4 × 10−4 nm−3 at 95 °C. This dif-ference in ordered state particle density translates to a ratio ofaverage particle spacing [(n/V)BCC/(n/V)C14]

1/3 = 1.18 that issimilar to the trend found in the corresponding disorderedfluids, q*(BCC,DIS)/q*(C14,DIS) = 1.21. Similarly, for the C15 phase(Fd�3m) we determine (n/V)C15 = 3.5 × 10−4 nm−3, and [(n/A)HEX]

1/2/[(n/V)C15]1/3 = 1.34, which is close to q*(HEX, DIS)/

q*(C15, DIS) = 1.30. In this case, we estimate the particle densityof the HEXC as n/A, where n is the number of cylindrical facesin a 2D hexagonal unit cell (i.e., n = 1 with a rhombic lattice asthe unit cell) and A is the area of the unit cell derived from theSAXS data (Table S2).

DiscussionThe results exposed by this work lead to the inescapable con-clusion that the disordered liquid state produced by particle-forming diblock copolymers in the limit of low molar mass, i.e.,as �N → 1, supports multiple discrete long-lived nonequilibriumstructures. These distinct metastable states, characterized bydifferent particle densities n/V, spawn different ordered struc-tures when cooled below TODT. We believe that all of the tem-peratures explored in this study are lower than TMDT andspeculate that the TMDT is well above the degradation temper-ature of the polymers and is therefore experimentally in-accessible. Unambiguous evidence of the structured nature ofthe disordered state has been presented previously (38–40).Linear dynamic mechanical spectroscopy measurements showthat the disordered liquid exhibits a plateau modulus in the dy-namic mechanical storage modulus G′ that is associated with adensely packed micellar state (5). Because these polymers havemolar masses well below the melt entanglement limit, there is noambiguity regarding the origins of this viscoelastic behavior. Atthe temperatures employed in this work we estimate that theterminal relaxation time associated with the disordered fluid

is well under 1 s (5), implying that the fluid is homogeneouson a macroscopic scale over the times associated with theSAXS experiments. However, the ordered Frank–Kasperphases, including the C14 and C15 Laves phases, are notspatially uniform. Instead, these low-symmetry crystals con-tain discrete distributions of particle sizes and shapes, dic-tated by the constraints imposed by incompressibility and thedriving force to fill space at essentially uniform density. As aconsequence, the soft particles assume shapes and local co-ordinations that reflect the polyhedral cells associated with thespecific lattice symmetry. Accessing these low-symmetry crystalstructures requires a combination of conformational asymmetryand/or molar mass dispersity as revealed by the SCFT calculationshighlighted in Fig. 2. Transitions between ordered states, and withthe disordered state, are mediated by the exchange of diblockcopolymer chains to create the required mix of particle volumes.Fig. 5 illustrates the structural changes which we speculate

accompany order and disorder in such diblock copolymer melts.Below TODT the distribution of particle shapes and sizes hasbeen unambiguously established as described in the previoussection. Upon heating, disordering necessitates a redistributionof particle volumes and shapes so as to avoid thermodynamicallyunfavorable variations in local density. We have shown earlier(5) that the required mass transfer becomes activated at tem-peratures above what is referred to as the ergodicity temperatureTerg << TODT. Hence, we anticipate facile chain exchange in thevicinity of TODT, consistent with rapid melting. We believe re-distribution of diblock copolymer molecules reshapes the parti-cle size distribution in the disordered state to the monomodalform illustrated in Fig. 5.However, chain exchange alone will not modify n/V. There is a

free energy barrier to varying the particle density, which requireseither particle fusion or particle fission as illustrated in Fig. 6.Well-established principles regarding the self-assembly ofdiblock copolymers show that there is a severe thermodynamicpenalty for doubling or halving the particle volume (47–49),which is required to change n/V based on two-body interactions.This barrier will become increasingly severe as the diblock co-polymer molar mass is reduced (while simultaneously increasingthe magnitude of χ so as to keep TODT at an experimentallytractable value). As �N is reduced from infinity, the value of χNwhere the strong segregation limit (SSL) sets in (loosely defined

A B

Fig. 4. Small-angle X-ray scattering (SAXS) results obtained from IL-52-20 (Mn = 4,830 g/mol, fL = 0.20, Ð = 1.07). (A) Specimen was freeze-dried and heated to140 °C, followed by cooling to 100 °C. Subsequent heating and cooling to these temperatures produced scattering patterns consistent with states of disorder andHEXC symmetry. (B) A freeze-dried specimen was heated to 140 °C then immersed in liquid nitrogen and reheated to 100 °C. Heating to 140 °C and cooling to100 °C produced SAXS patterns that correspond to a disordered fluid and the Laves C15 phase, respectively. Data are shifted vertically for clarity. These SAXSresults were duplicated during multiple heating and cooling cycles. Vertical marks identify the predicted peak positions for the ordered states (35).

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as the point of saturation of the composition profile at nearlypure block concentrations) declines from roughly (χN)SSL ≈100 to (χN)SSL << (χN)ODT as �N → 1 (50–52). This accounts forthe strong deviation from unperturbed Gaussian chain statistics(53) and mean-field behavior in the disordered state at all

compositions as �N → 1, as documented experimentally (28, 54–58)and captured by theory (57–60). In the asymmetric limit, thestructured disordered state occupies a broad window (TMDT >>TODT) as illustrated in Fig. 1. Whereas symmetric diblocks (fA = 1/2)can adjust the length scale of the disordered fluctuating bicontin-uous (and the ordered lamellar) morphology by diffusion of chainsalong the microdomain interface (55), the discontinuous particlesthat form in the asymmetric limit require the aforementionedfusion/fission events to change particle density. Hence, asym-metric diblocks are fundamentally different dynamically fromcompositionally symmetric ones. In the symmetric limit, small-angle scattering and rheological measurements have estab-lished a bona fide picture of the consequences of fluctuations(55). The situation in the compositionally asymmetric limit isless understood. Recently, a fluctuation corrected version of themean-field theory for sphere-forming AB diblocks has beenreported (61). However, a detailed theoretical understanding ofthe disordered phase is still lacking, especially in terms of itsmicellar nature.Reducing �N traps the disordered liquid in a state with a specific n/

V dictated by the processing history, i.e., freeze drying or quenchingin liquid nitrogen and reheating. The nonequilibrium particle den-sity, shaped by processing, can assume a wide range of values, whichthen guide the formation of specific crystal symmetries when cooled.We believe these concepts account for the memory effect displayedby the PI-PLA materials. Moreover, we conjecture that the expan-sive design space afforded by blending diblocks, either before orafter being situated in the strongly fluctuating disordered state, willprovide access to additional complex crystalline phases.SCFT does not predict the large variation in n/V that is found

experimentally between the BCC and C14 (or HEXC and C15)phases (Fig. S8). We hypothesize that the lower micelle populationsin the C14 and C15 phases are achieved by thermodynamic in-stability when the sample is quenched in liquid nitrogen andreheated. Quenching the sample in liquid nitrogen drives thesystem far from equilibrium and traps the configuration of thedisordered state. Upon heating, as soon as the sample escapesTg (or Terg, below which the system behaves like a glass) it might

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2a

V BCC

Fig. 5. Schematic of the proposed local structures in the ordered and disordered states obtained with IL-58-15 deduced from the SAXS results. A singleparticle size and shape is associated with the BCC phase, which has one particle per lattice site. Heating results in a distribution of particle sizes through chainexchange with an average volume ⟨V⟩BCC dictated by the particle density established in the ordered state. Cooling returns the material to a BCC crystal. TheC14 phase is characterized by 12 particles per unit cell, distributed between three discrete types of particle sizes and shapes, with an average volume ⟨V⟩C14.Heating above TODT leads to a redistribution of mass resulting in a monomodal distribution of particle sizes while maintaining the particle density formedduring initial formation of the C14 phase. The Laves phase returns when the material is cooled below TODT.

Fig. 6. Proposed mechanism for explaining the trapping of a fixed particledensity in the disordered state. Fusion or fission of pairs of micelles, whichrequires chain stretching and compression, respectively, away from thepreferred configurations, are kinetically constrained by large free-energy (F)barriers. Facile and rapid chain exchange provides for a continuous re-distribution of diblock copolymer molecules when T > Terg, which accom-modates the small variations in particle volume, ΔV = ±eVi, required to fillspace at constant density in the liquid state. Fluctuations in the averageparticle volume are indicated by the dashed circles at the ratio of initial tofinal number of particles per unit volume ni/nf = 1.

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lie on an unstable point on the free energy surface resulting inthe spontaneous redistribution of chains, similar to spinodaldecomposition. Such an instability-driven transformation couldentail rapid micelle fusion and fission (which is impossible toachieve via slow heating and cooling as mentioned above) asthe specimen is heated, producing a specific number densitywhich remains constant on further heating within the orderedregime and leads to the formation of the C14 and C15 phases.Note that there would be some free chains in the system above

the TODT, but we speculate that the number is small near the TODTin the disordered state [the free chain volume fraction is estimatedto increase by less than 5% upon BCC–micellar liquid transition(38)]. Samples were annealed for 5–15 min before cooling the meltsbelow the TODT, which is orders of magnitude longer than the re-laxation times (<1 s) of the short PI-PLA diblock copolymersstudied in this work (5). We expect that preserving the melt abovethe TODT for several hours, or perhaps many days, could result inan equilibrium distribution of micelles, with density, shapes, andsizes that may be different from those reported here.We anticipate that the consequences of these speculated mech-

anisms may be far reaching. Many other soft materials that assembleinto mesomorphic particles should be subject to the same phe-nomena. Particularly relevant are surfactant/water/oil systemssuch as those reported by Mahanthappa and coworkers (16, 18,19), who have demonstrated strikingly similar phase behavior tothat displayed by asymmetric diblock copolymer melts. Surfactantsrepresent the ultimate �N → 1 limit, while offering a plethora ofpractical applications made possible by control over the amphiphi-licity based on nonionic and ionic chemical structures. In principle,these arguments are scale invariant, suggesting possible approachesto controlling order and disorder at larger length scales.

Concluding RemarksSoft and hard materials constructed from approximately spher-ical constituents display surprisingly similar low-symmetry or-dered Frank–Kasper and quasicrystalline phases when cooled fromthe liquid state. We have suggested elsewhere that a commonfeature denoted “sphericity” underpins this apparent universality(5). However, there are fundamental differences between metallicand micellar liquids that mediate symmetry breaking upon cooling.Whereas metal atoms are immutable particles with fixed sizes(excluding the valence electrons that fill the conduction band),diblock copolymer micelles can assume a distribution of sizes andshapes, essential features in filling space at uniform density. Crys-tallization of an elemental metal or alloy requires rearranging thepositions of the constituent atoms at the Ångström length scale andreconfiguring the density of electronic states to accommodate thelattice symmetry, thus maximizing the average sphericity of the Joneszone in reciprocal space (62). These events are not constrained bydiffusion limitations as evidenced by facile phase transitions betweencrystalline phases in metals such as iron, manganese, and plutonium.Conversely, ordering a structured diblock copolymer liquid necessitatesredistribution of molecules between micelles to optimize sphericity atconstant density, which is a diffusion-controlled process. Fluctuationeffects in self-assembled diblock copolymers create conditions in theliquid state that do not exist in liquid metals: a manifold of long-livedmetastable states that trap the micelle number density at valuesdetermined by the processing conditions. Consequently, order-ing and symmetry breaking of disordered asymmetric diblockcopolymers is intimately connected to the liquid state structureand inextricably coupled to the effects of fluctuations (63), whichaffect the properties of various types of complex fluids (64–66).We speculate the fluctuating disordered state traps a type of hiddensymmetry that is manifested when the liquid is cooled below TODT,bearing semblance to other hidden symmetries commonly found inthe context of spontaneous symmetry breaking (67), including thoseoccurring in elementary particles (68) and spin glasses (69).

The combined experimental and theoretical results uncoveredby this work raise significant challenges in determining the trueequilibrium state of asymmetric diblock copolymers. As notedabove, it should be possible to narrow the fluctuating window inFig. 1 between TODT and TMDT by increasing the molar mass,which may extinguish the memory effect observed here becausethe liquid state will become homogeneous after heating a fewdegrees above TODT. This affords enticing opportunities tocomprehensively assess the role of fluctuation effects, both ex-perimentally and theoretically, in the fascinating region of orderand disorder that lies between the realms of asymptotically largeamphiphilic molecules ( �N →∞) and low-molar-mass compounds(�N → 1) such as water.

Dr. Breed was mistaken about at least one thing: there was such athing as ice-nine.

Chapter 23, Cat’s Cradle by Kurt Vonnegut (1)

Materials and MethodsSCFT Calculations. SCFT is a mean-field theory in which the local monomerinteractions aremodeled by the Flory–Huggins interaction parameter χ and theconfiguration of the chains are modeled using the standard Gaussian model(53). We employ unit cell SCFT calculations, in which the governing equationsare solved numerically within one unit cell, constraining the symmetry of thestructure under consideration (41, 43). The obtained solution then yields thefree energy of a stress-free unit cell of the ordered structure, representing alocal minimum in the free energy surface. To construct the phase diagram, theunit cell SCFT calculations for all of the candidate phases are performed, andthe free energies of the different structures are compared throughout theparameter space to determine the stable and metastable phases in a canonicalensemble. In the grand canonical ensemble, which is convenient for analyzingblends, the quantity of interest is the chemical potential of each phase.

In thiswork,we considered a binary blendof diblock copolymers, A1B andA2B, asshown in Fig. 2A. The system is composed of n1 chains of A1B and n2 chains of A2Bin a volumeV, such that ϕ1= n1N1/(n1N1+ n2N2) is the volume fraction of A1B in theblend, where N1 =NA1 +NB and N2 =NA2 +NB are the degrees of polymerizationfor the two diblocks, respectively. The blended melt is incompressible, and both thediblocks are conformationally symmetric, i.e., bA= bB. The SCFT parameter space forthis blend system thus consists of five parameters: ϕ1, χN1,NA1=N1,NA2=N1, and N2/N1. Here we examined the phase behavior only in the N2/N1–ϕ1 plane and for aparticular case, χN1 = 40, NA1 = 0.15N1, and NB = 0.85N1, considering the followingcandidate ordered structures: the Frank–Kasper C14, C15, σ, and A15 phases, alongwith BCC and HEXC. Furthermore, we used a combination of canonical and grand-canonical SCFT calculations to efficiently determine the regions of phase coexistencesuch as the one depicted by “two-phase” in Fig. 2B. The details of the calculationsincluding the governing equations for both canonical and grand-canonical en-semble are reported in the supporting information of Liu et al. (37).

Synthesis and Structural Examination of PI-PLA Diblock Copolymer Melts.Synthetic details of the PI-PLA polymers are described elsewhere (35, 55,70). Briefly, 1,4-polyisoprene homopolymer with a hydroxyl end group wassynthesized via anionic polymerization of isoprene monomer. This homo-polymer was used as a macroinitiator to polymerize (±)-lactide, generatingthe PI-PLA diblock copolymer. The freeze-dried samples (20–30 mg each)were hermetically sealed in aluminum DSC pans and subjected to variousthermal treatments: (i) directly quenching the samples from T > TODT totarget annealing temperatures (below the TODT) or (ii) immersing the samplesfrom T > TODT in a liquid nitrogen bath, then reheating the sample to targetT < TODT. The samples were annealed at the target temperatures for up to 20 dto form the documented morphologies (i.e., BCC, HEXC, or the Laves phases)(35). To study the temperature-dependent morphological behavior, SAXSexperiments were performed at Beamlines 5-ID-D DND-CAT and 12-ID-B atthe Advanced Photon Source at Argonne National Laboratory. Beam energiesand sample-to-detector distances were 17 keV/8.5 m at 5-ID-D and 12 keV/2.3 m at 12-ID-B, respectively. After confirming the formation of each orderedphase, the samples were heated to T > TODT with a heating rate of ∼2 °C/susing a Linkam DSC600 sample stage. The samples were then cooled back to thetarget temperature at a moderate cooling rate (i.e., ∼2 °C/s). The samples werethen held at the final temperature for up to 24 h to allow reordering. Theheating–cooling cycle was repeated up to four times to check consistency. Morethan five individual DSC pans were tested to confirm reproducibility. Data with

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well-delineated Bragg peaks were analyzed to determine crystallographic spacegroup and unit cell dimensions (71).

ACKNOWLEDGMENTS. Byeongdu Lee is acknowledged for his help withexperiments conducted at Beamline 12-ID-B at the Advanced Photon Source(APS). Marc Hillmyer provided a valuable review of the manuscript. Thisresearch was supported by the National Science Foundation under GrantsDMR-1104368 and DMR-1333669, the National Science Foundation of Chinaunder Grants 21774025 and 21320102005, and the Natural Science and Engi-neering Research Council of Canada. The work was enabled by resources at the

APS, a US Department of Energy (DOE) Office of Science User Facility operatedfor the DOE Office of Science by Argonne National Laboratory under ContractDE-AC02-06CH11357. Part of this work was performed at the DuPont–North-western–Dow Collaborative Access Team (DND-CAT) located at Sector 5 ofthe APS. DND-CAT is supported by E.I. DuPont de Nemours & Co., The DowChemical Company, and Northwestern University. Parts of the computa-tions were made possible by the Minnesota Supercomputing Institute atUniversity of Minnesota, as well as the facilities of the Shared HierarchicalAcademic Research Computing Network (www.sharcnet.ca) and Compute/Calcul Canada.

1. Vonnegut K (1963) Cat’s Cradle (Dell Publishing, New York).2. Callister WD, Rethwisch DG (2014) Materials Science and Engineering: An Introduction

(John Wiley & Sons, Inc., Hoboken, NJ), 9th Ed.3. Hobbs D, Hafner J, Spišák D (2003) Understanding the complex metallic element Mn. I.

Crystalline and noncollinear magnetic structure of α-Mn. Phys Rev B 68:014407.4. Hafner J, Hobbs D (2003) Understanding the complex metallic elementMn. II. Geometric

frustration in β-Mn, phase stability, and phase transitions. Phys Rev B 68:014408.5. Lee S, Leighton C, Bates FS (2014) Sphericity and symmetry breaking in the formation of Frank-

Kasper phases from one component materials. Proc Natl Acad Sci USA 111:17723–17731.6. Hecker SS (2000) Plutonium and its alloys–From atoms to microstructure. Los Alamos

Sci 26:290–335.7. De Graef M, McHenry ME (2012) Structure of Materials: An Introduction to

Crystallography, Diffraction, and Symmetry (Cambridge Univ Press, New York), 2nd Ed.8. Shechtman D, Blech I, Gratias D, Cahn JW (1984) Metallic phase with long-range

orientational order and no translational symmetry. Phys Rev Lett 53:1951–1953.9. Levine D, Steinhardt PJ (1984) Quasicrystals: A new class of ordered structures. Phys

Rev Lett 53:2477–2480.10. Bartels-Rausch T, et al. (2012) Ice structures, patterns, and processes: A view across the

icefields. Rev Mod Phys 84:885–944.11. Falenty A, Hansen TC, Kuhs WF (2014) Formation and properties of ice XVI obtained

by emptying a type sII clathrate hydrate. Nature 516:231–233.12. Meldrum FC, Cölfen H (2008) Controlling mineral morphologies and structures in

biological and synthetic systems. Chem Rev 108:4332–4432.13. Seddon JM (1990) An inverse face-centered cubic phase formed by diacylglycerol-

phosphatidylcholine mixtures. Biochemistry 29:7997–8002.14. Seddon JM, Robins J, Gulik-Krzywicki T, Delacroix H (2000) Inverse micellar phases of

phospholipids and glycolipids. Phys Chem Chem Phys 2:4485–4493.15. Pouzot M, et al. (2007) Structural and rheological investigation of Fd�3m inverse mi-

cellar cubic phases. Langmuir 23:9618–9628.16. Perroni DV, Mahanthappa MK (2013) Inverse Pm3̄n cubic micellar lyotropic phases

from zwitterionic triazolium gemini surfactants. Soft Matter 9:7919–7922.17. Sorenson GP, Schmitt AK, Mahanthappa MK (2014) Discovery of a tetracontinuous,

aqueous lyotropic network phase with unusual 3D-hexagonal symmetry. Soft Matter10:8229–8235.

18. Sorenson GP, Mahanthappa MK (2016) Unexpected role of linker position on am-monium gemini surfactant lyotropic gyroid phase stability. Soft Matter 12:2408–2415.

19. Kim SA, Jeong K-J, Yethiraj A, Mahanthappa MK (2017) Low-symmetry sphere packings ofsimple surfactant micelles induced by ionic sphericity. Proc Natl Acad Sci USA 114:4072–4077.

20. Hudson SD, et al. (1997) Direct visualization of individual cylindrical and sphericalsupramolecular dendrimers. Science 278:449–452.

21. Ungar G, Liu Y, Zeng X, Percec V, Cho W-D (2003) Giant supramolecular liquid crystallattice. Science 299:1208–1211.

22. Zeng X, et al. (2004) Supramolecular dendritic liquid quasicrystals. Nature 428:157–160.23. Garcia-Bennett AE, Miyasaka K, Terasaki O, Che S (2004) Structural solution of mes-

ocaged material AMS-8. Chem Mater 16:3597–3605.24. Huang M, et al. (2015) Self-assembly. Selective assemblies of giant tetrahedra via

precisely controlled positional interactions. Science 348:424–428.25. Yue K, et al. (2016) Geometry induced sequence of nanoscale Frank-Kasper and qua-

sicrystal mesophases in giant surfactants. Proc Natl Acad Sci USA 113:14195–14200.26. Feng X, et al. (2017) Hierarchical self-organization of ABn dendron-like molecules into

a supramolecular lattice sequence. ACS Cent Sci 3:860–867.27. Bates FS, Fredrickson GH (1990) Block copolymer thermodynamics: Theory and ex-

periment. Annu Rev Phys Chem 41:525–557.28. Bates FS, et al. (1994) Fluctuations, conformational asymmetry and block copolymer

phase behaviour. Faraday Discuss 98:7–18.29. Bates FS, Fredrickson GH (1999) Block copolymers—Designer soft materials. Phys

Today 52:32–38.30. Mai Y, Eisenberg A (2012) Self-assembly of block copolymers. Chem Soc Rev 41:5969–5985.31. Doerk GS, Yager KG (2017) Beyond native block copolymer morphologies. Mol Syst

Des Eng 2:518–538.32. Leibler L (1980) Theory of microphase separation in block copolymers. Macromolecules

13:1602–1617.33. Lee S, Bluemle MJ, Bates FS (2010) Discovery of a Frank-Kasper σ phase in sphere-

forming block copolymer melts. Science 330:349–353.34. Gillard TM, Lee S, Bates FS (2016) Dodecagonal quasicrystalline order in a diblock

copolymer melt. Proc Natl Acad Sci USA 113:5167–5172.35. Kim K, et al. (2017) Thermal processing of diblock copolymer melts mimics metallurgy.

Science 356:520–523.36. Xie N, Li W, Qiu F, Shi A-C (2014) σ phase formed in conformationally asymmetric AB-

type block copolymers. ACS Macro Lett 3:906–910.37. Liu M, Qiang Y, Li W, Qiu F, Shi A-C (2016) Stabilizing the Frank-Kasper phases via

binary blends of AB diblock copolymers. ACS Macro Lett 5:1167–1171.

38. Dormidontova EE, Lodge TP (2001) The order−disorder transition and the disorderedmicelle regime in sphere-forming block copolymer melts.Macromolecules 34:9143–9155.

39. Wang X, Dormidontova EE, Lodge TP (2002) The order−disorder transition and thedisordered micelle regime for poly(ethylenepropylene-b-dimethylsiloxane) spheres.Macromolecules 35:9687–9697.

40. Wang J, Wang Z-G, Yang Y (2005) Nature of disordered micelles in sphere-formingblock copolymer melts. Macromolecules 38:1979–1988.

41. Arora A, et al. (2016) Broadly accessible self-consistent field theory for block polymermaterials discovery. Macromolecules 49:4675–4690.

42. Schulze MW, et al. (2017) Conformational asymmetry and quasicrystal approximantsin linear diblock copolymers. Phys Rev Lett 118:207801.

43. Matsen MW, Schick M (1994) Stable and unstable phases of a diblock copolymer melt.Phys Rev Lett 72:2660–2663.

44. Hamley IW, et al. (1993) Hexagonal mesophases between lamellae and cylinders in adiblock copolymer melt. Macromolecules 26:5959–5970.

45. Hajduk DA, et al. (1994) Observation of a reversible thermotropic order–order tran-sition in a diblock copolymer. Macromolecules 27:490–501.

46. Zhao J, et al. (1996) Phase behavior of pure diblocks and binary diblock blends ofpoly(ethylene)−poly(ethylethylene). Macromolecules 29:1204–1215.

47. Izzo D, Marques CM (1993) Formation of micelles of diblock and triblock copolymersin a selective solvent. Macromolecules 26:7189–7194.

48. Dormidontova EE (1999) Micellization kinetics in block copolymer solutions: Scalingmodel. Macromolecules 32:7630–7644.

49. Hadgiivanova R, Diamant H, Andelman D (2011) Kinetics of surfactant micellization: Afree energy approach. J Phys Chem B 115:7268–7280.

50. Matsen MW, Whitmore MD (1996) Accurate diblock copolymer phase boundaries atstrong segregations. J Chem Phys 105:9698–9701.

51. Matsen MW, Bates FS (1996) Unifying weak- and strong-segregation block copolymertheories. Macromolecules 29:1091–1098.

52. Matsen MW (2010) Strong-segregation limit of the self-consistent field theory fordiblock copolymer melts. Eur Phys J E Soft Matter 33:297–306.

53. Matsen MW (2002) The standard Gaussian model for block copolymer melts. J PhysCondens Matter 14:21–47.

54. Bates FS, Rosedale JH, Fredrickson GH (1990) Fluctuation effects in a symmetric di-block copolymer near the order–disorder transition. J Chem Phys 92:6255–6270.

55. Lee S, Gillard TM, Bates FS (2013) Fluctuations, order, and disorder in short diblockcopolymers. AIChE J 59:3502–3513.

56. Hickey RJ, Gillard TM, Lodge TP, Bates FS (2015) Influence of composition fluctuationson the linear viscoelastic properties of symmetric diblock copolymers near the order–disorder transition. ACS Macro Lett 4:260–265.

57. Guenza M, Schweizer KS (1997) Fluctuations effects in diblock copolymer fluids:Comparison of theories and experiment. J Chem Phys 106:7391–7410.

58. Gillard TM, Medapuram P, Morse DC, Bates FS (2015) Fluctuations, phase transitions,and latent heat in short diblock copolymers: Comparison of experiment, simulation,and theory. Macromolecules 48:2801–2811.

59. Fredrickson GH, Helfand E (1987) Fluctuation effects in the theory of microphaseseparation in block copolymers. J Chem Phys 87:697–705.

60. Guenza M, Schweizer KS (1997) Local and microdomain concentration fluctuationeffects in block copolymer solutions. Macromolecules 30:4205–4219.

61. Delaney KT, Fredrickson GH (2016) Recent developments in fully fluctuating field-theoretic simulations of polymer melts and solutions. J Phys Chem B 120:7615–7634.

62. Lee S, et al. (2013) Pseudo-fivefold diffraction symmetries in tetrahedral packing.Chemistry 19:10244–10270.

63. Tanaka H (2013) Importance of many-body orientational correlations in the physicaldescription of liquids. Faraday Discuss 167:9–76.

64. Murata K, Tanaka H (2015) Microscopic identification of the order parameter governingliquid-liquid transition in a molecular liquid. Proc Natl Acad Sci USA 112:5956–5961.

65. Kleiman RN, Bishop DJ, Pindak R, Taborek P (1984) Shear modulus and specific heat ofthe liquid-crystal blue phases. Phys Rev Lett 53:2137–2140.

66. Hickey RJ, Gillard TM, Irwin MT, Lodge TP, Bates FS (2016) Structure, viscoelasticity, and in-terfacial dynamics of a model polymeric bicontinuous microemulsion. Soft Matter 12:53–66.

67. Gross DJ (1996) The role of symmetry in fundamental physics. Proc Natl Acad Sci USA93:14256–14259.

68. Seiberg N, Witten E (1994) Monopoles, duality and chiral symmetry breaking in N =2 supersymmetric QCD. Nucl Phys B 431:484–550.

69. Newman CM, Stein DL (2003) Ordering and broken symmetry in short-ranged spinglasses. J Phys Condens Matter 15:R1319–R1364.

70. Schmidt SC, Hillmyer MA (1999) Synthesis and characterization of model polyisoprene−polylactide diblock copolymers. Macromolecules 32:4794–4801.

71. Hahn T (2005) International Tables for Crystallography: Vol. A. Space-Group Symmetry(Springer, Norwell, MA), 5th Ed.

854 | www.pnas.org/cgi/doi/10.1073/pnas.1717850115 Kim et al.

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