Discovery of a Frank-Kasper σ Phase in Sphere-Forming Block Copolymer Melts

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Science  15 Oct 2010:
Vol. 330, Issue 6002, pp. 349-353
DOI: 10.1126/science.1195552


Sphere-forming block copolymers are known to self-assemble into body-centered cubic crystals near the order-disorder transition temperature. Small-angle x-ray scattering and transmission electron microscopy experiments on diblock and tetrablock copolymer melts have revealed an equilibrium phase characterized by a large tetragonal unit cell containing 30 microphase-separated spheres. This structure, referred to as the sigma (σ) phase by Frank and Kasper more than 50 years ago, nucleates and grows from the body-centered cubic phase similar to its occurrence in metal alloys and is a crystal approximant to dodecagonal quasicrystals. Formation of the σ phase in undiluted linear block copolymers (and certain branched dendrimers) appears to be mediated by macromolecular packing frustration, an entropic contribution to the interparticle interactions that control the sphere-packing geometry.

The organization of atoms, molecules, and larger aggregates into condensed phases with specific symmetries and spatial dimensions dictates the physical properties of all materials. Naïvely, spherical particles might be expected to present the simplest case. Yet filling space in three dimensions with spherical objects can produce surprisingly complex structures, including disordered liquids, vitrified glasses, crystalline solids, and quasicrystals. Rare gas atoms, governed by attractive van der Waals interactions and strongly repulsive cores, such as argon and xenon, behave as simple hard spheres and form either face-centered cubic (fcc) or hexagonal close-packed (hcp) crystals at low temperatures and high pressures (1). These close-packed structures, along with body-centered cubic (bcc) packing, account for all but a few of the metallic elements in the periodic table (2). Colloidal crystals present a similar packing problem but at much larger length scales, typically 0.1 to 1 μm. Modern synthetic and processing techniques permit the preparation of uniformly sized spherical colloidal particles that can be arranged as fcc, hcp, or bcc crystals, depending on the surface treatment and processing strategies employed (35).

Many soft materials contain spherical molecular assemblies at mesoscopic length scales, intermediate to those associated with atomic and colloidal crystallization, including lipids and surfactants (6), dendrimers (7) and block copolymers (8). Ordering in these systems often reflects a tendency to minimize packing frustration, that is, elimination of entropically expensive molecular configurations necessitated by the incompressible nature of soft matter (9). These constraints are particularly evident in block copolymers (10), which exhibit a host of self-assembled morphologies depending on the molecular architecture (e.g., AB diblock and ABC triblock), block chemistry and composition, and overall molecular weight. Thirty years ago, Leibler anticipated that near the order-to-disorder transition (ODT), asymmetric diblock copolymer melts would form spherical microdomains ordered on a bcc lattice (11), a prediction that is now universally accepted (12).

Leibler also considered the possibility of icosahedral order in block copolymer melts, motivated in part by a provocative publication by Alexander and McTague describing a general treatment of crystal melting based on Landau theory (13). However, because objects with five-fold symmetry can not form periodic structures with translational order, this option was dismissed by both groups. Discovery of quasicrystalline order in multicomponent alloys in 1984 (14, 15), just 4 years after Leibler’s seminal contribution, heralded a new era in material design, one that continues to expand today. Hundreds of aperiodically ordered compounds have been reported with underlying 5-, 7-, 8-, 10- and 12-fold symmetry, usually generated by rapidly supercooling a multicomponent liquid (16). With one noteworthy exception (17), we are unaware of any experimental realizations of quasicrystals formed from single-component, sphere-forming liquids, although computer simulations suggest that this may be possible.

Keys and Glotzer (18) reported the growth of a metastable dodecagonal quasicrystal phase in molecular simulations of a supercooled liquid of spherical particles governed by an interatomic potential due to Dzugutov (19), known to produce local icosahedral clustering. The stable (equilibrium) state for this system is believed to be the σ phase (20), a crystal structure formed from coordinated polyhedra as first described by Frank and Kasper (21) more than 50 years ago. There is a close relationship between aperiodic order and periodic crystals with large unit cells; the Frank-Kasper σ phase represents the “periodic approximant” to certain dodecagonal quasicrystals (18).

Soft materials are likely candidates for the formation of Frank-Kasper phases and the corresponding quasicrystals (2224). A tendency to minimize packing frustration should favor coordination shells among spherical particles that may be incompatible with cubic and hexagonal crystallization for certain molecular architectures. We are aware of only one example of a single-component, sphere-forming soft material (and one class of nonspherical two-component blends) (25) that exhibits such phase behavior. Ungar and co-workers (26) reported the σ and bcc phases with increasing temperature, and subsequently a dodecagonal quasicrystal (17), in wedge-shaped dendrimer molecules that self-assemble into approximately spherical particles. They argue that these morphologies, and a litany of other structures (7), are controlled by molecular shape, dictated by specific branched architectures.

Block copolymers offer nearly unlimited flexibility for the design of microphase-separated soft materials based on chain statistics common to all linear molecular architectures. In this report, we demonstrate formation of the Frank-Kasper σ phase in two different single-component, sphere-forming, block copolymer melts near the order-disorder transition temperature (TODT). This finding anticipates fresh opportunities for designing soft materials with new and potentially useful ordered, and possibly quasicrystalline, structures.

Two undiluted linear block copolymers were investigated. They were prepared by sequential anionic polymerization and characterized using well-established techniques (27, 28). IL-15 is a poly(isoprene-b-lactide) (PI-PLA) diblock copolymer with a number-average molecular weight Mn = 3.89 kg/mol and containing 28 weight percent PLA and polydispersity index Mw/Mn = 1.12. Based on the corresponding homopolymer densities, this compound contains 22 ± 1% by volume PLA blocks. Differential scanning calorimetry measurements establish glass transition temperatures of –66°C (PI) and 5°C (PLA) for this material (fig. S1). SISO-3 is a poly(styrene-b-isoprene-b-styrene-b-ethylene oxide) (PS-PI-PS-PEO) tetrablock copolymer with a number-average molecular weight Mn = 23 kg/mol and polydispersity index Mw/Mn = 1.04, containing 46% by volume PS (divided equally), 46% PI and 8% PEO (each ± 1%); synthesis and characterization of this compound has been described elsewhere (28). This material forms PEO-rich spheres dispersed in a mixed PS and PI matrix above the melting (T ≈ 38°C) and glass transition (T < 100°C) temperatures, with TODT,SISO-3 = 224 ± 3°C. Both molecular structures are illustrated in Fig. 1.

Fig. 1

Molecular architecture of poly(isoprene-b-lactide) (IL) and poly(styrene-b-isoprene-b-styrene-b-ethylene oxide) (SISO).

Small-angle x-ray scattering (SAXS) experiments were conducted at the Advanced Photon Source (Argonne National Laboratory) to assess the state of order in these block copolymers, employing a sample-to-detector distance of 4.04 or 6.64 m and radiation wavelength λ = 0.729 Å. Scattered x-rays were recorded on a Mar CCD area detector and azimuthally averaged to the one-dimensional form of intensity (I) versus scattering wave vector magnitude |q|=q=4πλ1sin(θ/2), where θ is the scattering angle. Figure 2A shows a series of SAXS measurements obtained after rapidly cooling (≈ 100°C/min) sample IL-15 from 70°C to 40°C. Several minutes after cooling, the broad peak associated with disordered state (correlation hole scattering) spawns sharp, instrument-resolution limited, diffraction peaks that saturate after about 30 min with relative positions q/q*=1,2,3,4,5,6,7, where q* = 0.0648 Å−1 is the location of the lowest-order reflection. This diffraction pattern is consistent with bcc symmetry (unit cell size a = 137 Å), which we associate with the self-assembly of PLA rich spheres in a PI matrix. Based on the molecular composition, the radius of these spheres is R = 41 Å, which is consistent with form-factor scattering at higher q (not shown). Each sphere incorporates approximately 193 block copolymer molecules.

Fig. 2

Synchrotron SAXS powder patterns obtained from sample IL-15. (A) After cooling from 70 to 40°C. (B) After cooling from 120 to 25°C. Growth of bcc crystals at 40°C is evidenced by the development of Bragg peaks at Embedded Image. Initial development of cubic order at 25°C (Embedded Image) is replaced by a slowly evolving state of order that derives from the σ phase.

Three-dimensional ordering transforms disordered block copolymer melts into soft solids accompanied by a large increase in the dynamic elastic shear modulus G′ (29). We monitored G′ at a frequency of 0.1 rad/s using a mechanical spectrometer after rapidly cooling IL-15 from 120° to 40°C (fig. S2). This experiment confirms that nucleation and growth of the cubic phase takes about 30 min. Upon heating the bcc ordered material from 40°C, the diffraction pattern disappears and G′ collapses at 50 ± 1°C, which is identified as TODT,IL-15.

A qualitatively different SAXS powder pattern develops when sample IL-15 is rapidly cooled from T > TODT,IL-15 to 25°C, as illustrated in Fig. 2B. Initially, scattering peaks consistent with the bcc structure appear, but these are replaced by a set of completely different reflections over the course of about 1 day. After aging this specimen at room temperature (~25°C) for 26 days, we recorded a remarkable diffraction pattern, containing at least 48 distinct peaks as shown in Fig. 3A, which cannot be indexed based on any previously reported block copolymer morphology. Dynamic elasticity measurements reveal a two-step ordering process at 25°C. Evolution of G′ (0.1 rad/s) during the first 30 min after a quench from 120°C follows the behavior documented at 40°C, then the shear modulus continues to slowly grow, by an additional 50% over the next 24-hour period (fig. S2).

Fig. 3

Synchrotron SAXS powder patterns from (A) IL-15 after 26 days at room temperature, and (B) SISO-3 after 1 day at 140°C. The q axes have been scaled to facilitate comparison of the relative positions and intensities of these scattering patterns, which are both associated with the σ phase. A model (Rietveld) simulation of the data in (A) for 0.017 ≤ q ≤ 0.081 Å−1 based on space group P42/mnm is presented in (C).

We obtained the same unusual SAXS pattern shown in Fig. 3A from sample SISO-3 between 140° and 224°C (TODT), as illustrated in Fig. 3B (140°C); these data were taken after 1 day of annealing at 140°C (30). To facilitate comparison of the data taken from IL-15 and SISO-3, we have adjusted the q scales in Fig. 3 so that the peaks align. Although the SAXS pattern from SISO-3 contains fewer high q reflections, the relative positions and intensities of at least 15 peaks nearly duplicate those recorded with IL-15, leading us to conclude that these materials contain the same structure. Discovery of these nearly identical SAXS results was serendipitous but helped to convince us that this complicated scattering pattern was indeed characteristic of a well-defined ordered morphology.

The positions of the diffraction peaks in Fig. 3, A and B, are consistent with tetragonal symmetry yielding unit cell dimensions: a = 431 Å, c = 228 Å for IL-15 (indexing is provided in table S4), and a = 777 Å and c = 411 Å for SISO-3. Assuming identical spherical morphologies, each large unit cell would contain 30.5 domains based on the initial bcc (Fig. 2B, 4.7 hours) and final (Fig. 3A) Bragg scattering from IL-15. Ungar et al. (26) reported a SAXS pattern for undiluted wedge-shaped dendrimers that nearly duplicates the 10 most intense peaks found in Fig. 3. They determined that these branched compounds assembled into approximately spherical particles (each containing on average 11.6 molecules) on a tetragonal lattice, 30 particles per unit cell, with P42/mnm space group symmetry. (P4¯n2 and P42nm symmetry are also feasible; the authors assumed the highest allowed symmetry.)

Definitive assignment of a nanoscale morphology to a material is best accomplished based on complementary reciprocal space (i.e., scattering) and real space (i.e., microscopy) analyses. Although IL-15 provides access to an impressive number of SAXS reflections, this specimen is not easily characterized by transmission electron microscopy (TEM) due to the combined effects of a low matrix molecular weight and a reduced glass transition temperature, which complicate microtoming and transferring of thin slices onto microscope grids. Fortunately, SISO-3 is sufficiently robust to permit sample preparation for TEM imaging. A specimen was annealed in vacuum at 140°C for 10 hours, then frozen by plunging into liquid nitrogen. After warming to room temperature, a small piece of the stiff glassy material was sectioned using a Reichert cryo-ultramicrotome. Thin slices (~70 nm thick) were transferred onto copper grids, stained by exposure to the vapor from a 4% aqueous osmium tetraoxide solution (OsO4 reacts selectively with the PI blocks), and examined with a FEI F30 instrument operated at 300 kV. We found sizable areas of well-ordered morphology in this material (up to 1 μm by 0.4 μm in size), as illustrated in Fig. 4.

Fig. 4

TEM image recorded from a stained (OsO4) thin section of sample SISO-3. The morphology is consistent with a tilted c-axis projection of the σ phase, where the unit cell is identified by the dashed rectangle. Isolated white spots surrounded by dark and gray rings are consistent with columns of dodecagonal units of spheres. A five-fold coordinated tiling element, formed from squares and triangles, is associated with each ring in the morphology, verifying the assignment of the σ phase to SISO-3.

Real-space electron density maps were generated from the IL-15 SAXS powder diffraction data using the Rietveld method with P42/mnm symmetry (space group 136) and a set of 30 spherical particles per unit cell. A background intensity function, modeled after the correlation hole scattering obtained immediately after a temperature quench to 25°C (Fig. 2B and fig. S3), was employed in this analysis. Comparison of the experimental and calculated I(q) patterns, determined for 0.017 ≤ q ≤ 0.081 Å−1, is shown in Fig. 3, A and C, respectively; optimized Wyckoff positions are listed in table S5. (Note that for q < 0.04 Å−1 the calculation is sensitive to minor variations in the scattering power assigned to each Wykoff position, which we estimated based on the Wigner-Seitz cell volumes, as described below. We believe the minor discrepancy between the computed and experimental scattering patterns in the low q range can be attributed to this effect). The resulting crystal structure is shown in Fig. 5A, and projection of a collection of unit cells along the c axis is presented in Fig. 5B. This complex periodic structure is formed from fused dodecagonal cells (layers of alternating hexagonally coordinated rings surrounding a column of spheres) resulting in a vertical projection that closely approximates the TEM image in Fig. 4. Owing to the finite thickness of the TEM specimen, we are not able to delineate every anticipated sphere in the micrograph. Nevertheless, an array of white spots surrounded by dark and gray rings, decorated with additional white spots, are plainly evident. Comparison of the calculated and experimental unit cells indicates that the TEM specimen was sliced at an oblique angle, distorting the four-fold tetragonal symmetry apparent in the calculated projection.

Fig. 5

(A) Unit cell for the σ-phase morphology (P42/mnm symmetry) based on a Rietveld analysis of the IL-15 SAXS powder pattern shown in Fig. 3. Columns of spheres (dashed blue lines) surrounded by fused rings of hexagonally coordinated spheres (solid blue lines) produce a distinctive pattern when projected along the c axis (B), which is consistent with the TEM image in Fig. 4. A tiling element, characteristic of the five-fold coordination of the dodecagonal elements in the σ phase, and the tetragonal face of the unit cell, are sketched on this image.

Frank and Kasper (21) postulated a rich assortment of space-filling algorithms in an effort to rationalize the structure of complex alloys formed from spherical particles. The σ phase, which has P42/mnm symmetry and 30 spheres per unit cell, is an important example of such a Frank-Kasper phase. We are aware of only two elemental manifestations of this structure, β-uranium (31) and β-tantalum (32), although the σ phase is common in metal alloys (33), particularly stainless steels, where it nucleates and grows in conjunction with bcc ferrite (34). A distinctive feature of this structure is the two-dimensional pattern created by tiling the four-fold plane perpendicular to the stacking (c) direction with triangles and squares. Figure 5B illustrates the basic tiling element of the σ phase: a unit containing two triangles, a square, a triangle, and a square. Five equal-length segments connect the nearest vertices of this element, identifying the five-fold coordination of dodecagonal stacks in the c-axis projection of the σ phase (Fig. 5B). After accounting for the aforementioned skewing, the TEM image can be tiled using the element (Fig. 4), corroborating our assignment of the Frank-Kasper σ phase to SISO-3 at elevated temperatures, and IL-15 at 25°C based on the SAXS identity in Fig. 3.

Ordered microphases in block copolymer melts reflect a delicate balance between enthalpic (H) and entropic (S) factors that govern the overall Gibbs free energy at a given temperature and pressure, G = H – TS. Self-assembly leads to ordered arrays of microdomains with specific interfacial area Σ ~ d2 (d is the characteristic domain dimension) and interfacial tension γ ~ χ1/2 (χ is the Flory-Huggins interaction parameter), where the product Σγ determines H. To maintain constant density at fixed domain geometry, the polymer blocks must adopt a distribution of molecular configurations that collectively define the overall entropy S. The equilibrium morphology and lattice parameters are determined by a tradeoff between minimizing H and maximizing S. Until now, the phases that characterize microphase-separated diblock copolymers (spheres, cylinders, gyroid, and lamellae) contain one uniform domain type, for example, one sphere per bcc lattice site (8). In contrast, five distinct sphere packing geometries, distributed between ten 12-fold, sixteen 14-fold, and four 15-fold coordinated sites characterize the σ phase.

We have constructed the five different Wigner-Seitz polyhedra that make up the overall volume of the σ-phase unit cell (fig. S6), ranging in volume from 91% to 106% of the average cell volume. This arrangement, which represents a compromise between the purely icosahedron and truncated octahedron (bcc) symmetries anticipated by Landau theory, appears to create additional packing frustration (i.e., a wider range of length scales that must be accommodated by stretched and compressed block configurations), thus reducing the system entropy. Occurrence of the bcc phase at higher temperatures than those associated with the σ phase in IL-15 is consistent with this entropic argument. Based on the lattice parameters obtained at 25°C before and after the bcc-to-σ transformation (Fig. 2B), the ratio of interfacial areas per unit volume is <Σσ>/Σbcc = 0.99. This implies a slightly smaller enthalpy for the σ phase relative to bcc, which would explain why it is favored over bcc packing at lower temperatures. Clearly, these simple observations must be tested with quantitative theory.

Although IL-15 and SISO-3 have relatively narrow molecular weight and composition distributions, these polymers are not strictly monomolecular and we cannot rule out the possibility that the polymer chains are distributed asymmetrically among the different sites within the σ-phase unit cell. However, the occurrence of this morphology in (presumably) pure monomolecular dendrimers argues against this entropically expensive explanation.

Finally, we are intrigued by the possibility of preparing soft quasicrystalline phases from single-component block copolymer melts. The σ phase is the approximant crystal structure to certain dodecagonal quasicrystals, and an example of both σ-phase and quasicrystal formation has been reported in a single-dendrimer compound (17, 26). Hayashida et al. also reported a dodecagonal quasicrystal in cylinder forming binary blends of star block terpolymer and homopolymer based on the nonperiodic tiling of TEM micrographs with nets (25). Molecular simulations indicate that dodecagonal quasicrystals represent slowly evolving metastable states relative to the (equilibrium) σ phase (18). Our time-dependent SAXS (Figs. 2 and 3) and mechanical spectroscopy (fig. S2) experiments with IL-15 may reflect a transient quasicrystalline morphology, consistent with these simulations. Appropriately designed block copolymers could represent ideal materials with which to characterize the thermodynamic and kinetic properties of these fascinating aperiodic systems.

Identification of the σ phase in linear block copolymer melts presents the opportunity for designing interesting and useful materials from many other polymers. Decades of experimental experience and well-established theory show that single-component block copolymer melts exhibit universal phase behavior, governed by well-established molecular parameters, primarily the molecular weight, composition, and segment-segment interaction parameter χ (8). The σ phase has an enormous unit cell, with lattice parameters that are expected to scale with the two-thirds power of block copolymer molecular weight. Thus, an asymmetric diblock copolymer prepared with 20 times the molecular weight of IL-15 (i.e., a modest 80 kg/mol), and subject to judicious choice of block types (which controls χ), should result in a unit cell dimension of a ≈ 0.3 μm, potentially suitable for application as a photonic crystal (35). More generally, the concept of guiding the growth of gigantic crystals and quasicrystals by tailoring the packing frustration of soft macromolecules might be extended to other organic and inorganic nanoparticles by attaching polymer chains with controlled molecular weight and polydispersity to the surfaces.

Supporting Online Material

Figs. S1 to S7


References and Notes

  1. The integrity of sample SISO-3 after 24 hours of annealing under vacuum at 140°C was verified by size exclusion chromatography (SEC).
  2. This work was supported by the Department of Energy through a subcontract to UT-Battelle (4000041622), the National Science Foundation through grant DMR- 0704192, and the University of Minnesota Materials Research Science and Engineering Center (MRSEC). Portions of this work were performed at the DuPont-Northwestern-Dow Collaborative Access Team (DND-CAT) located at Sector 5 of the Advanced Photon Source (APS). DND-CAT is supported by E. I. Dupont de Nemours & Company, the Dow Chemical Company, and the State of Illinois. Use of the APS was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under contract DE-AC02-06CH11357. Parts of this work were carried out in the University of Minnesota I.T. Characterization Facility, which receives partial support from NSF through the National Nanotechnology Infrastructure Network program. The authors gratefully acknowledge helpful discussions with N. Balsara and T. P. Lodge.
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