Introduction

The capability of spontaneously forming diverse periodic structures on nanometer scale makes block copolymer a fascinating class of soft matters, which has been extensively studied in the past a few decades and continues to draw tremendous attention nowadays1,2. It has been widely recognized that self-assembly behaviors of linear block copolymers follow an explicit and general principle that depends mainly on three variables: i.e., degree of polymerization (N), composition (f), and Flory−Huggins interaction parameter (χ)3,4. Classical phases, such as lamellae (LAM), bicontinuous double gyroids (DG), hexagonally packed cylinders (HEX), and body-centered cubic (BCC) packed spheres, can be readily obtained by tuning these molecular parameters4. Recent discoveries of exotic complex phases, including Frank−Kasper phases and quasicrystalline phases, greatly expand the diversity of the accessible nanostructures and sparks ever increasing interest in the field5,6,7,8,9.

Despite these exciting progresses, there is still a missing piece of the puzzle in the phase diagram of block copolymer. The canonical phase theories predict the existence of thermodynamically stable, close-packed spherical (CPS) phases, i.e., hexagonally close packing (HCP) and face centered cubic (FCC) packing, in a very narrow region near the order-disorder boundary4,10. Both are constructed by stacking two dimensional hexagonally close-packed layers of spheres, different only in the stacking sequences along the perpendicular direction (i.e., ABCABC stacking for FCC vs ABABAB stacking for HCP)11. Compared to the prevailing BCC phase, the Wigner−Seitz cells of HCP and FCC lattices are less spherical, thus mandating a severe non-uniform stretching of the corona block to fill the distal voids (Fig. 1)12. The emergence of the CPS lattices in the predicted phase diagram is attributed to the solubilization effect that a portion of short minority blocks dislodge from the core domain and swell the matrix as “homopolymer” additives, which effectively releases chain stretching of the corona block13,14. Unfortunately, composition fluctuation becomes dominant at these extreme compositions and destroys the long-range order of the close-packed lattices. As a result, CPS phases are only sporadically captured in experiments, exclusively in block copolymer/homopolymer blends or block copolymers with a certain chain length distribution11,15,16,17,18. In these cases, the homopolymer additives (or the block copolymers with an exceedingly short minority block) dissolve in the matrix and fill the far reaches of the Wigner−Seitz cells, releasing the overstretching of the corona blocks and increasing the conformational entropy13. Recently, Zhang and coworkers reported an elegant example of HCP structure in block copolymers, demonstrating that a broad molecular weight distribution could stabilize the close-packed phases18. Our group developed a modular approach to precisely regulate the chain length dispersity through blending discrete block copolymers of varying sizes, and confirmed that HCP phase would become thermodynamically stable when the effective distribution exceeds a critical threshold19. To the best of our knowledge, however, there has been no direct experimental observation of CPS phases in a strict single-component block copolymer, casting uncomfortable concerns regarding the fundamental theoretical principles developed from the ideal and uniform systems.

Fig. 1: Schematic illustration of spherical packings.
figure 1

a Wigner−Seitz cell of typical spherical lattices: FCC, HCP, and BCC. b Asymmetric triblock copolymers form the stable HCP phase through synergistic contribution of the shorter and longer end blocks. The bridging conformation is not shown for clarity.

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This study meticulously designed a series of isomeric linear A1BA2 triblock copolymers with a broken symmetry by varying the relative size of the end A blocks. This architecture introduces unique features that are not present in the AB diblock or symmetric ABA triblock counterparts20,21, circumventing the paradox that prevents the formation of CPS phases (Fig. 1b). The precise chemical composition and uniform chain length eliminate the inherent molecular weight distribution and other defects22. By tuning the molecular symmetry, a transition from HEX to A15, σ, and back to HEX was observed. Moreover, HCP lattice became thermodynamically stable attributed to the synergetic contribution of the two nonequivalent end blocks23. The longer blocks form the core domain and hold the ordered structure, whereas the shorter blocks are pulled out to release the stretching energy (Fig. 1b). Regulation of the molecular symmetry can effectively alter the free energy landscape, opening a robust pathway for modulating phase behaviors without changing the chemistry or composition and affording accesses to unconventional phases otherwise unattainable.

Results

Linear block copolymers Ln1DAmLn2 with a discrete number of repeat units were modularly prepared through the iterative exponential growth (IEG) approach with a combination of orthogonal protection/deprotection and esterification coupling reactions (n1, n2, and m are the numbers of repeat units; Fig. 2b, Supplementary Figs. 1–4, Supplementary Tables 1–4)24,25,26,27. The two end blocks are composed of oligo lactic acid (abbreviated as L, Supplementary Fig. 1 and Supplementary Table 1), and the middle block is composed of oligo 2-hydroxytetradecanoic acid (abbreviated as DA, Supplementary Fig. 2 and Supplementary Table 2); both are orthogonally protected by a tert-butyldimethylsilyl (TBDMS) and a benzyl (Bn) group28. The isomers have the same chemical composition but varying molecular symmetries. The size of the two end LA blocks can be finely adjusted, while the overall composition remains unchanged. For clarity, the TBDMS terminal is defined as the beginning of the block sequence. A unitless parameter τ = n1/(n1 + n2) was introduced to describe the symmetry of the triblock copolymers (Fig. 2a)23. The linear chains are symmetric triblock in the case τ = 0.5, while they are reduced to diblock copolymers when τ = 0. In this work, three sets of isomeric block copolymers were prepared, in which the L block is the minority component with a volume fraction (fL) between 0.23 and 0.31 (Table 1). The chemical structures of the block copolymers were characterized and confirmed by nuclear magnetic resonance (NMR), size exclusion chromatography (SEC), and matrix-assisted laser desorption ionization-time of flight mass spectrometry (MALDI-ToF MS) (Fig. 2c, d, Supplementary Figs. 5–16). For each species, a single peak was identified in the MS profile, substantiating the chemical structure and discrete feature. The isomers within each group have identical composition, making them an ideal system for elucidating the critical contribution of molecular symmetry, devoid of any undesirable interferences (Fig. 3)29. Detailed molecular information was summarized in Table 1.

Fig. 2: Design and characterization of discrete triblock copolymers.
figure 2

a Schematic illustration of triblock copolymers with varying symmetry. b Synthetic route of the linear triblock copolymers. c, d Representative SEC and MALDI-ToF MS profiles of L8DA24L24 and the related precursors.

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Table 1 Characterization of discrete triblock copolymers with varied architectural symmetry
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Fig. 3: MALDI-ToF MS of the triblock copolymer isomers with varying molecular symmetry.
figure 3

a n1 + n2 = 48; (b) n1 + n2 = 40; (c) n1 + n2 = 32.

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Despite their similar chemical features, the two blocks are immiscible due to the variation in their side chains (Fig. 2b). The glass transition temperatures of both blocks are below room temperature, resulting in rapid assembly kinetics (Supplementary Fig. 17). All block copolymers were briefly heated to a high temperature (ca. 150 °C) under a nitrogen atmosphere to remove any pre-existing thermal history and then annealed at room temperature to promote the formation of ordered nanostructures. Synchrotron small-angle X-ray scattering (SAXS) was utilized to study the phase behaviors of these block polymers.

The symmetric triblock copolymers with τ = 0.5 were first studied as a reference. The sample L16DA24L16 with the lowest volume fraction in this study (fL = 0.23) remains disordered at room temperature (Fig. 4c), due to insufficient segregation strength. On the other hand, L20DA24L20 with two longer L end blocks (fL = 0.27) forms a HEX structure with L cylinders embedding in the DA matrix, as indicated by a q/q* ratio of 1: √3: 2 from the scattering profile (Fig. 4b). Similar HEX lattice was identified in the sample L24DA24L24 (fL = 0.31, Fig. 4a), with a larger intercolumnar distance (d = 9.13 nm) compared to that of L20DA24L20 (d = 8.04 nm).

Fig. 4: SAXS profiles of discrete triblock copolymers with varying molecular symmetry.
figure 4

a n1 + n2 = 48; (b) n1 + n2 = 40; (c) n1 + n2 = 32. Representative spherical packings are shown in (d). The indexes can be found in supplementary discussions and calculations. Data are shifted vertically for clarity.

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Interestingly, simply adjusting the relative chain length of two end blocks leads to substantially different phase behaviors. For example, compared with the symmetric triblock copolymer L24DA24L24, the isomeric L16DA24L32 with a slightly different end block size (τ = 0.33) adopts a totally different packing scheme (Fig. 4a). A group of scattering peaks with a q ratio of √2: √4: √5: √6: √8: √10 was observed in the reduced 1D SAXS profile, which can be assigned to a Frank−Kasper A15 phase with a cubic lattice8. Further enlarging the asymmetry (L8DA24L40; τ = 0.17) triggers another phase transition, wherein the characteristic peaks of A15 lattice were replaced by a forest of sharp diffractions corresponding to a Frank−Kasper σ phase with P42/mnm symmetry (Fig. 4a; detailed peak indexing can be found in Supplementary Fig. 18)6. The lattice parameters of the tetragonal unit cell were calculated based on the principal scatterings (a = b = 46.61 nm and c = 24.68 nm). The increasing asymmetry also leads to an appreciable change in the size of the spherical motifs, with an average diameter increasing from <D > = 11.95 nm (for L16DA24L32) to 15.06 nm (for L8DA24L40; Table 1). On the other hand, when n1 further decreases to 0, the triblock copolymer is reduced to a simple diblock isomer DA24L48. The SAXS pattern indicates an interesting re-entry feature that the sample adopt the same HEX structure as the symmetric ABA counterpart, while the lattice dimension significantly increases to 13.24 nm. Overall, the linear block copolymers show rich phase structures with a transition from HEX to A15, σ, and back to HEX by tuning the relative size of two end blocks, without changing the chemistry or composition. Frank−Kasper phases are a fascinating class of low-symmetry nanostructures consisting of two or more types of spherical motifs with different sizes, shapes, and coordination numbers (Fig. 4d)9. While the conformational asymmetry between the two blocks is widely acknowledged for promoting the formation of these exotic phases, the presence of nonequivalent end blocks also plays an essential role in distinguishing them from conventional AB diblock or symmetric ABA triblock counterparts (see discussion below)30. A similar transition sequence was observed in another set of isomers L20DA24L20 (fL = 0.27; Fig. 4b).

More interestingly, in an asymmetric triblock copolymer with a lower volume fraction of oLA blocks (n1 + n2 = 32), an HCP phase was captured, as shown in Fig. 4c. While L16DA24L16 and L12DA24L20 remain disordered at room temperature due to insufficient segregation strength, a diffraction pattern consisting of three prominent intense diffraction peaks, together with several weak diffractions, emerges in the SAXS profile of L8DA24L24 (τ = 0.25; Fig. 4c). It perfectly matches an HCP spherical lattice, in which the three strong peaks can be assigned as (010), (002), and (011), respectively11,18. The calculated lattice parameters a = b = 10.71 nm and c = 17.48 nm well fit to the characteristic ratio of HCP lattice (c/a = 1.63). The average diameter <D> of the spherical motif was calculated to be 11.83 nm. More surprisingly, no coexistence of FCC packing was observed, as confirmed by the complete absence of the {200} family of reflections (Supplementary Fig. 18d). A small fraction of stacking faults might exist, as suggested by the relatively broader (012) peak31,32. Given nearly degenerate free energies, it is rather surprising to capture a pure FCC or HCP structure, even though SCFT calculations indicate the latter is slightly favored33. More importantly, the observation of an equilibrium HCP phase in single-component block copolymers challenges the conventional wisdom that the CPS phase is difficult to obtain in linear block copolymer melts without the use of blending or other complex processing techniques11,18. On the other hand, the sample DA24L32 with τ = 0 self-assembles into a Frank-Kasper σ structure (Supplementary Fig. 19). The average diameter of the spherical motifs also increases to <D > = 13.55 nm.

The stability of the HCP phase was further examined through in-situ SAXS experiments (Fig. 5). Sample L8DA24L24 was slowly heated on a hot stage mounted on the SAXS apparatus. The HCP structure remains stable at low temperatures, while the characteristic peaks began to merge at 35 °C (Fig. 5a). A new set of scatterings emerged at 40 °C with a q/q* ratio of 1: √2: √3: √4, indicating a phase transition to a BCC lattice (lattice parameter a = 12.16 nm). The spherical motifs in the BCC lattice have a similar diameter (<D > = 11.97 nm) as that of HCP lattice (11.83 nm), indicating this transition mainly involves rearrangement of the spherical motifs in the lattice, while the micelles remain intact. Upon further heating, the diffraction peaks turned into a broad bump at 60 °C and the system entered a disordered state (Fig. 5a). In the subsequent cooling cycle, the BCC structure first emerged and then transformed into HCP lattice at a lower temperature (Fig. 5b). The reversibility of the order-to-order transition (OOT) confirms that HCP structure is an equilibrium phase (Fig. 5c). On the other hand, the σ structure of DA24L32 persists at a much higher temperature, entering the disordered state at 180 °C (Supplementary Fig. 19).

Fig. 5: Reversible transitions between BCC and HCP lattices.
figure 5

In-situ heating (a) and cooling (b) processes of sample L8DA24L24 reveal reversible transitions between BCC and HCP lattices (c). Temperature and annealing time are listed in the figure.

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Temperature dependent SAXS experiments were conducted to capture additional information on phase formation and transition (Supplementary Figs. 19–21). The stability of the assembled structures was found to be highly influenced by the molecular symmetry. For example, a significant increase of order-to-disorder transition temperature (TODT) from 90 °C of L24DA24L24 (τ = 0.5) and 85 °C of L16DA24L32 (τ = 0.33) to 140 °C of L8DA24L40 (τ = 0.17), and to 280 °C of DA24L48 (τ = 0) was observed in the sample series with a total n = 48 (Supplementary Fig. 21). Similar behaviors were observed in other triblock copolymers with different volume fractions (Supplementary Fig. 20).

Discussion

Upon cooling below the order-to-disorder transition temperature, the block copolymers consisting of incompatible chains first aggregate into spherical micelles, which further pack into periodic lattices with optimal symmetry. During this process, the local preference to maintain the native shape of the assembled motifs and the global constraint to uniformly fill the space cannot be simultaneously optimized, generating packing frustration34,35. The spherical motifs are thus deformed into faceted polyhedra (i.e., Wigner−Seitz cells) with the shape imposed by the lattice symmetry (Fig. 1a). As a result, polymer chains are nonuniformly stretched to fill the Wigner−Seitz cells. The equilibrium structure is dictated by a delicate competition between the interfacial tension (enthalpic term) and the stretching of the polymer chains (entropic term). In general, the lattice consisting of Wigner−Seitz cells with the least deviation from the spherical geometry is usually favored12. Since the truncated octahedron cell of the BCC lattice has a higher sphericity than the trapezo-rhombic dodecahedron cell of HCP and rhombic dodecahedron cell of FCC, block copolymers are packed predominantly into the BCC phase (Fig. 1a). On the other hand, recent studies have also revealed that a large conformational asymmetry between two blocks (and thus a stiffer corona) leads to a strong coupling between the shape of the core/corona interface and the shape of the Wigner−Seitz cell, driving the formation of Frank−Kasper phases36,37. Though the exact segment length of DA block is not available, it can be roughly estimated based on a similar monomer, γ-dodecyl-α-hydroxy glutaric acid (DGA, b ≈ 0.43 nm)38. The conformational asymmetry between DA and L blocks can thus be estimated to be ε = bL/bD ≈ 1.84, which is well above the limit for the formation of these unconventional spherical phases8,37.

Blending homopolymers or introducing molecular weight distribution has been demonstrated as an effective way to alleviate the packing frustration through local segregation of the longer and shorter blocks within the motif or uneven partition among different motifs30,39,40,41,42,43. Asymmetric linear triblock copolymers with “built-in” chain length heterogeneity provide an excellent alternative to effectively alter the free energy landscape44,45. Compared to the symmetric triblock isomer, the longer end blocks can extend to the center of the core, while the short end blocks mainly occupy the interfacial area (i.e., shell). This core-shell structure was supported by the segmental density distribution calculated based on a self-consistent field theory (SCFT) (see Supplementary Information for detailed discussions; Supplementary Fig. 25)30,39. The synergies reduce the excessive chain stretching, which is conducive to generating large and deformable polyhedral motifs. As a result, introducing a slight asymmetry in the triblock further promotes the formation of the Frank−Kasper phases (Fig. 4a, b). The observation is also in line with the recent SCFT computational work on the phase behaviors of the block copolymer blends30. The optimal accommodation of the longer and shorter chains in a limited space determines the lattice selection.

The corona chains in the FCC/HCP lattices suffer from severe stretching to reach the distal interstices, accompanied with a large entropic loss that destabilizes the close packing lattice. In the triblock copolymers A1BA2 with sufficiently large asymmetry, the shorter end block is no longer firmly tied to the core domain. Instead, it will be partially pulled into the matrix, forming a dangling “middle block + ” and filling the distal region of the Wigner−Seitz cell (Fig. 1b)18,23. This rationale was substantiated by the SCFT calculations, in which a significant portion of short A1 block was found to disperse in the matrix region (Supplementary Fig. 25). Though unfavorable interactions occur when an A block is forced to contact with B matrix, this is more than compensated for by allowing the B block to relax. The longer A block, on the other hand, forms a stable spherical core, holding the periodic structure against fluctuation. The synergies between the shorter and longer end blocks can effectively release packing frustration, circumventing the dilemma that prevents the formation of CPS lattice. When the asymmetry of the triblock copolymer reaches an extreme (i.e., diblock copolymer), the synergistic effect no longer exists, resulting in the closure of the HCP window. To clearly demonstrate the differences, an additional collection of diblock copolymers DA24-Ln with varying n (n = 8, 16, 20, and 24) were prepared (Supplementary Table 4). In the diblock system, a transition sequence from HEX to σ, and to BCC was captured before entering the disordered state, with the HCP lattice being notably absent (Supplementary Fig. 22). Although the possibility of a narrow HCP window cannot be completely excluded (most likely between BCC and disordered region, Supplementary Fig. 22a), these apparent differences clearly underscore the crucial role of chain architecture in determining the lattice selection.

Different from hard sphere systems in which FCC is the most preferred close-packed structure46, a relatively higher thermodynamic stability of HCP was predicted in block copolymer systems by a later SCFT calculation33. While the overall packing fraction and cell sphericity are identical, distribution of the interstitial voids in the space is not. There exist two different types of voids, i.e., octahedral voids (larger) and tetrahedral voids (smaller)47. In HCP lattice, octahedral voids share common faces with a direct route from one to another, whereas the octahedral voids are interrupted by tetrahedral voids in FCC lattice32. The local interstitial environment of the former thus offers better conformational freedom to the corona chains, outweighing the positional entropy gain in FCC lattice. It should be noted that, however, considering vanishingly small free energy difference (~ 10−4 kBT per chain), whether FCC or HCP is the more stable CPS phase in block copolymers is still an interesting open question33.

In summary, this study showcases a robust method for regulating the phase behavior of linear block copolymers by rationally breaking molecular symmetry and marks a direct experimental observation of the HCP spherical lattice in a rigorous single-component block copolymer system. The synergies between the shorter and the longer end blocks significantly alter the free energy landscape, providing access to rich unconventional structures without changing the chemistry or composition. This observation unequivocally confirms the presence of an equilibrium HCP phase in single-component block copolymers, validating the theoretical predictions and filling in the missing pieces of the puzzle.

Methods

Syntheses of discrete Ln1DAmLn2 block copolymers

With the Ln1DAm and DAmLn2 precursors, a library of isomers was modularly constructed through esterification (Supplementary Fig. 4). Here we take L8DA24L24 as an example. Other copolymers were obtained following the same procedure.

General procedure for TBDMS-Ln1DAm-COOH

TBDMS-L8DA12-Bn (600 mg, 0.170 mmol, 1.0 eq) was dissolved in ethyl acetate (50 mL). Palladium (10% on carbon, 100 mg) was then added. The mixture was stirred at room temperature under a hydrogen atmosphere for 5 h. The black suspension was filtered through a thick layer of kieselguhr, and the filter cake was washed with EA (3 × 50 mL). After removing solvent, the product (TBDMS-L8DA12-COOH) was obtained as a colorless oil (580 mg, 99% yield) and used directly.

General procedure for HO-DAmLn2-Bn

TBDMS-DA12L24-Bn (250 mg, 0.054 mmol, 1.0 eq) was dissolved in 5 mL of anhydrous CH2Cl2 in an ice bath. BF3-etherate (22 mg, 0.162 mmol, 3 eq) was then slowly dropped into the solution. The mixture was allowed to return to room temperature and was further stirred for 4 h. After the reaction was completed (as monitored by TLC), it was quenched by pouring into a saturated aqueous solution of NaHCO3 (2 mL). The resulting organic phase was washed with deionized water (3 × 2 mL) and dried with anhydrous Na2SO4. The crude product was purified by recycling preparative SEC, yielding the product HO-DA12L24-Bn as a colorless oil.

General procedure for Ln1DAmLn2

TBDMS-L8DA12-COOH (83 mg, 0.024 mmol, 1.1 eq), HO-DA12L24-Bn (100 mg, 0.022 mmol, 1.0 eq), and DPTS (10 mg) were dissolved in 1 mL of dried DCM. DIC (4 mg, 0.036 mmol, 1.5 eq) was then slowly dropped into the above solution with an ice bath. The mixture was stirred at room temperature overnight. The crude product was purified by recycling preparative SEC, yielding the product TBDMS-L8DA24L24-Bn as a colorless oil (18 mg, 10% yield). 1H-NMR (CDCl3, 500 MHz): δ 7.38-7.29 (m, 5H), 5.36-4.96 (m, 57H), 4.40 (m, 1H), 2.05-1.74 (br, 48H), 1.62-1.52 (m, 93H), 1.50-1.18 (m, 483H), 0.88 (m, 81H), 0.08 (p, 6H). MS (MALDI-TOF, m/z): [M+Na]+ Cal 7983.47; Obs 7982.98.

Structural characterization

Small angle X-ray scattering (SAXS) data were collected on Shanghai Synchrotron Radiation (SSRF), beamline BL16B1. The incident X-ray wavelength (λ) was 0.124 nm (photon energy: 10 keV; photo flux: 1 × 1011 phs/s). The beam size was around 0.4 × 0.5 mm2. Scattered X-rays were captured on a 2-dimensional Pilatus detector. The scattering vector (q) was calibrated using a standard of silver behenate. In-situ experiments were carried out with a Linkam hot stage mounted onto the SAXS apparatus. Samples were sealed with aluminum foil for good thermal conductivity. The heating rate was 10 °C/min. Samples were equilibrated for 3 min at each temperature before collecting data.