Complex self-assembled lattices from simple polymer blends

Block copolymers having two or more polymer chains linked covalently have been extensively studied for various nanostructure fabrications. Highly symmetric body-centered cubic (bcc) packing lattice has been the only observed supramolecular spherical phase in compositionally asymmetric AB diblock copolymers for a long time. After 2010, the Frank−Kasper (F-K) σ (1) and A15 (2) phases featured by small volume asymmetry among constitutional supramolecular motifs have been discovered in some block copolymers, although with elaborately designed block conformation asymmetry or architectures. In PNAS, Cheong et al. (3) utilize the self-consistent field theory (SCFT) to demonstrate that thermodynamically stable F-K Laves phases with intrinsically large volume asymmetry can be realized by simply blending diblock copolymers with minority-block homopolymer in dry-brush mode. This rather convenient strategy potentially provides common block copolymer systems the access to those complex condensed phases with large volume asymmetry and upper length scale.

F-K phases in soft matter are a family of complex spherical phases sharing similar lattice structures with some metallic alloys. They are tetragonally close-packed structures, with spherical motifs deformed into polyhedra to accommodate the lattice symmetry and maintain constant density. These polyhedra are combinations of coordination number 12 (CN12) polyhedron, CN14 polyhedron, CN15 polyhedron, or CN16 polyhedron (4). Discovery of F-K phases in soft matter can be traced back to the 1990s, and their observation in various soft matter systems and the deeper understanding of them has accelerated in recent years. So far, F-K phases have been identified in four main categories of soft matter, including small molecular surfactants (5, 6), dendrimers (7, 8), block copolymers (1, 2, 9), and giant amphiphiles (10–12). A distinct feature of F-K phases in soft matter is that the constructing polyhedra are assembled from multiple, identical molecular constituents. These polyhedral, scaling from several to tens of nanometers, spontaneously form volume partition asymmetry during self-assembly driven by the balance between A/B interfacial tension and polymer chain stretching (1). This kind of spontaneous symmetry breaking of volume partition is, of course, limited by the entropy penalty due to polymer chain stretching. In most single-component systems, F-K A15 and σ phases with relatively small volume asymmetry among polyhedra in constructing Voronoi cells (Fig. 1A) are more frequently observed. F-K phases with enhanced volume asymmetry, such as Laves C14 phase (relative volumes between 113.9% and 92.8%), Laves C15 phase (relative volumes of 114.2% and 92.2%), and Z phase (112.6% and 90.1%), are only sporadically observed. Particularly, the C14 and C15 phases reported by Kim et al. in the superfast cooling of conformation-asymmetric diblock copolymer melts were demonstrated to be metastable phases (9). Moreover, either complicated branched architecture (13–15) or high conformational asymmetry [ε = (b_A/b_B)², where b is the segment length of polymer] condition (1, 2, 9, 16) is necessary to induce the F-K phase formation in block copolymers (Fig. 1C), and specific chemical structure design and tedious synthesis steps are usually required. How to make these unconventional phase structures become conventional in common linear diblock copolymers is rather challenging.

Fig. 1.

(A) The lattice structures of F-K A15, σ, Z, C14, and C15 phases, and corresponding relative volumes of constructing polyhedra with respect to average volume of all polyhedra in the corresponding phase. CN12, CN14, CN15, and CN16 polyhedra are presented by red, blue, green, and yellow polyhedra, respectively. (B) The schematic illustration of the AB block copolymer/A’ homopolymer blends in the dry-brush model. After adding the homopolymer, a volume-symmetric bcc phase transforms into a volume-asymmetric C15 phase. (C) Road map of the average diameter of deformed spherical motifs in F-K phases among different soft matter systems.

Very recently, Mueller et al. (17) reported an experimental emergence of F-K C14 and C15 phases via simply blending poly(styrene-block-1,4-butadiene) (AB, ε = 1.7) with poly(1,4-butadiene) (A) homopolymer (same repeating units with the minority block). Although the blending of block copolymers/homopolymers dates back to the early 1970s, detailed studies of the spherical phases arising from swelling sphere-forming diblock cores with a homopolymer have been scarcely reported. This work is of particular interest to the polymer society, since this convenient approach could be general without the requirement of either large conformational asymmetry or the specific branched architecture of the block copolymers. However, the distribution of homopolymer cannot be accessed experimentally, which is significant for understanding the mechanism of forming those complex spherical phases. Now, a simulation work based on SCFT published in PNAS by Cheong et al. (3) not only qualitatively reproduces the salient features of the experimental phase behaviors but also provides a clear picture of the homopolymer partitioning particle by particle in these complex phases with large volume asymmetry. Based on the simulated results, the homopolymer swelling of minority block core in a dry-brush mechanism is confirmed.

In an AB/A (A is the minority block) blending system, besides the χN (where χ is the Flory−Huggins parameter and N is the degree of polymerization of the diblock polymer), volume fraction of A block in diblock copolymer ϕ_A, and volume fraction of homopolymer ϕ_H, one of the most important factors in determining the phase behavior of the blends is the ratio α = N_A,homo/N_A,block (N_A,homo is the chain length of A-homopolymer; N_A,block is the chain length of A block in AB diblock copolymer). For α < 1, the A-homopolymer uniformly swells the A microdomain, as a so-called “wet-brush” model, driving an increase of the A/B interfacial area and, consequently, a decrease of the average curvature of the interface. For α > 1, macrophase separation between AB diblock copolymers and A-homopolymer matrix would occur. For α ≈ 1, a blend adapts a “dry-brush” model, wherein the A-homopolymer is strongly excluded from the A/B interphase and located at the center of the A domain core. In the “dry-brush” model, the interfacial area per diblock chain does not change even after the addition of A-homopolymer. In the work of Cheong et al. (3), the blending system (χN = 25, α = 1) was calculated to exhibit a phase sequence of bcc, σ, C14, and C15 phases with increasing volume of homopolymers. With the continuous addition of A-homopolymer, the volume-symmetric bcc phase will finally transform to F-K C15 phase (Fig. 1B), which has the highest polyhedron volume asymmetry phase in block copolymer systems so far. The effect of partitioning homopolymer to the position at the center of the polyhedron is revealed to be much stronger in F-K Laves phase than in initial bcc phase. The average homopolymer volume fraction at the center of the polyhedral increases significantly from 0.3 in bcc to around 0.5 in σ, 0.7 in C14, and 0.9 in C15 phase. Furthermore, for a given F-K phase, the homopolymer volume fraction at the center of the polyhedron monotonically increases with the coordination number of the polyhedra, which themselves have increased volume with larger coordination number.

Decreasing the length of homopolymer additive gives smaller α values (7/9 and 2/3), driving the systems into the “wet-brush” regime. The volume fraction of homopolymer at the center of the polyhedron gets lower as the α value becomes smaller. Consequently, the windows for the F-K phases get narrower as the brush gets “wetter.” As the volume fraction of homopolymer further increases, the hexagonal columnar phase appears, due to the reduced mean curvature of the interface.

In addition to the extremely simple chemical structures of block copolymers, another advantage of this “dry-brush” blending approach is worth mentioning. F-K phases in block copolymers usually require the adoption of multiple polyhedron volumes and thus greater stretching or compressing of polymer chains. Therefore, F-K packing structures are usually observed in low-molecular-weight diblock copolymers (Mw < 10 kg/mol), accompanied by low length scale of constructing polyhedra. Notably, the length scale is even lower in other soft matter systems. In Fig. 1C, the length scales of average polyhedral motifs in different soft matter systems are compared. Obviously, the average diameters of deformed spherical motifs (or polyhedra) are less than 20 nm in dendrimers, giant amphiphiles, lipid surfactants, and most single-component block copolymers. This presents a limitation for the potential application of these nanostructures in further study. The SCFT calculation by Cheong et al. (3) and experiments by Mueller et al. (17) reveal that these F-K phases can be realized in block copolymers with a relatively large degree of polymerization by “dry-brush” homopolymer blending. Besides, the “dry-brush” homopolymer swelling of the minority core provides additional expansion of the spherical motifs, about 25%. The final size of individual spherical motif can achieve 31 nm in F-K C15 structure. Interestingly, another method of blending two AB diblock copolymers with different chain lengths was also demonstrated to facilitate F-K A15 and σ phase formation and push the average diameter of constructing polyhedra to around 35 nm (18).

The diblock copolymer/homopolymer blending in the “dry-brush” range is fully elucidated by SCFT analysis in this work and corroborated as a general approach to construct complex spherical phases with large volume asymmetry. An important but not an easy task in further exploration is how to precisely control the volume asymmetry among the deformed spherical motifs for lattice structure tuning. In other words, could the degree of spontaneous volume asymmetry and subsequent packing structure be predicted by SCFT to help rationalize specific polymer structure design before tedious synthesis? One step further, could the limit of volume asymmetry degree be further pushed to seek other unprecedented metal alloy analog phases in polymers, not only constrained to the F-K phases? For example, decagonal quasicrystals with 10-fold rotational symmetry are often observed in the phase transition of metal alloys with Laves C15 structure (19). In metallurgy, decagonal quasicrystal has been demonstrated to be a thermodynamically stable structure when a third element with larger atom radius is added (20). The powerful SCFT could be an efficient tool to directly explore these complex phases.