Among the various ordered morphologies self-assembled from block copolymers, the spherical packing phases are particularly interesting because they resemble the familiar atomic crystals. The commonly observed spherical morphology of block copolymers is the body-centered-cubic phase. Recently, a number of novel spherical packing phases, i.e., the complex Frank–Kasper phases originally obtained in metallic alloys, have been observed in block copolymer melts. Theoretical studies have revealed that conformational asymmetry of the different blocks provides a key mechanism to stabilize the Frank–Kasper phases. Furthermore, local segregation of different copolymers in blends of diblock copolymers and copolymer architectures provides additional mechanisms to enhance the stability of the complex ordered phases. In this Viewpoint we summarize recent advances in our understanding of the formation of the nonclassical spherical packing phases in AB-type block copolymers, emphasizing the formation mechanisms of these fascinating complex ordered structures.

Monolayers of linear and miktoarm star ABC triblock copolymers with equal A and C blocks were investigated using self-consistent field theory. Monolayers of ABC triblock copolymers were formed between two parallel surfaces that were attractive to the A and C blocks. The repulsive interaction parameter χACN between the A and C blocks was chosen to be weaker than the A/B and B/C interactions, quantified by χABN and χBCN, respectively, such that the B blocks were confined at the A/C interface, resulting in various B domains with different geometries and arrangements. It was observed that two variables, namely, the strength of the surface fields and the film thickness, were dominant factors controlling the self-assembly of the B blocks into various morphologies. For the linear triblock copolymers, the morphologies of the B domains included disks, stripes (parallel cylinders), and hexagonal networks (inverse disks). For the miktoarm star triblock copolymers, the competition between the tendency to align the junction points along a straight line and the constraint on their arrangement from the surface interactions led to richer ordered morphologies. As a result of the packing of the junction points of the ABC miktoarm star copolymers, a counterintuitive phase sequence from low-curvature phases to high-curvature phases with increasing length of B block was predicted. The study indicates that the self-assembly of monolayers of ABC triblock copolymers provides an interesting platform for engineering novel morphologies.

The phase behavior of binary blends composed of AB diblock and (A′B)n star copolymers is studied using the polymeric self-consistent field theory, focusing on the formation and stability of the stable tetragonal phase of cylinders. In general, cylindrical domains self-assembled from AB-type block copolymers are packed into a hexagonal array, although a tetragonal array of cylinders could be more favourable for lithography applications in microelectronics. The polymer blends are designed such that there is an attractive interaction between the A and A′ blocks, which increases the compatibility between the two copolymers and thus suppresses the macroscopic phase separation of the blends. With an appropriate choice of system parameters, a considerable stability window for the targeted tetragonal phase is identified in the blends. Importantly, the transition mechanism between the hexagonal and tetragonal phases is elucidated by examining the distribution of the two types of copolymers in the unit cell of the structure. The results reveal that the short (A′B)n star copolymers are preferentially located in the bonding area connecting two neighboring domains in order to reduce extra stretching, whereas the long AB diblock copolymers are extended to further space of the unit cell.

Patterning strategies based on directed self-assembly (DSA) of block copolymers, as one of the most appealing next-generation lithography techniques, have attracted abiding interest. DSA aims at fabricating defect-free geometrically simple patterns on large scales or irregular device-oriented structures. Successful application of DSA requires to control and optimize multiple process parameters related to the bulk morphology of the block copolymer, its interaction with the chemical or topographical guiding pattern, and the kinetics of structure formation. Most studies have focused on validating DSA patterning techniques using PS-b-PMMA block copolymers as a prototypical material. As the development of DSA techniques advances, recent efforts have been devoted to extending the materials selection in order to fabricate more complex geometric patterns or patterns with smaller characteristic dimensions. How to select appropriate polymer materials in a vast parameter space is a critical but also challenging step. In this review, we discuss recent progress in the research of DSA of block copolymers focusing on three aspects: (i) screening the block copolymer materials, (ii) controlling the film properties, and (iii) tailoring the phase separation kinetics.

The emergence of the complex Frank-Kasper phases from binary mixtures of AB diblock copolymers is studied using the self-consistent field theory. The relative stability of different ordered phases, including the Frank-Kasper σ and A15 phases containing nonspherical minority domains with different sizes, is examined by a comparison of their free energy. The resulting phase diagrams reveal that the σ phase occupies a large region in the phase space of the system. The formation mechanism of the σ phase is elucidated by the distribution of the two diblock copolymers with different lengths and compositions. In particular, the segregation of the two types of copolymers, occurring among different domains and within each domain, provides a mechanism to regulate the size and shape of the minority domains, thus enhancing the stability of the Frank-Kasper phases. These findings provide insight into understanding the formation of the Frank-Kasper phases in soft matter systems and a simple route to obtain complex ordered phases using block copolymer blends.

The heterogeneous nucleation process during the phase separation of binary blends of the AB diblock and the C homopolymer induced by rectangular confinement is studied by cell dynamics simulation based on the time-dependent Ginzburg–Landau theory. The main goal is to yield large-scale ordered hexagonal patterns by tailoring the surface potentials of the sidewalls. Our study reveals a crucial condition to induce the desired heterogeneous nucleation process in which the nucleated domain grains grow and merge into a defect-free pattern. Specifically, nucleations are induced simultaneously by two parallel sidewalls with a strong surface potential, whereas the spontaneous nucleation and the heterogeneous nucleation at the other two walls with a weak surface potential are suppressed. Moreover, the confinement effect of the other two walls can ensure that the two rows of nucleated domains have correlated positions. Importantly, we find that the ordering process under the crucial condition exhibits a high tolerance to the rectangular sizes. Only a few defects in thousands of domains are occasionally caused that are observed to be annihilated in a short-annealing time via various mechanisms. This study may provide a facile route to prepare large-scale ordered patterns via a simple rectangular confinement.

The ordering dynamics of cylinder-forming diblock copolymer/homopolymer blends confined in hexagonal potential wells is systematically investigated using time-dependent Ginzburg–Landau (TDGL) theory. It is demonstrated that a high-efficient method to obtain large-scale ordered hexagonal patterns is to utilize corner-induced heterogeneous nucleation processes, in which nucleation events with controlled positions and orientations are triggered exclusively at the six corners of the confining hexagonal wells. Subsequent growth of the six domains originated from the corners leads to the formation of perfectly ordered patterns occupying the entire hexagonal well. The heterogeneous nucleation rate is regulated by the homopolymer concentration as well as the surface potential of the confining walls. Defect-free hexagonal patterns are obtained in hexagons with a diagonal size containing up to 61 cylinders (about 2 μm). The robustness of the method is examined by studying the tolerance window of the size-commensurability of the confining wells. The results indicate that controlled heterogeneous nucleation provides an efficient method for the fabrication of large-scale ordered patterns using graphoepitaxy of block copolymer self-assembly.

The phase behavior of binary blends composed of BABCB-type pentablock terpolymers and ABC-type triblock copolymers is investigated using the self-consistent field theory in the grand canonical ensemble. Specifically, the study is focused on how the simpler copolymers regulate the phase behavior of two sphere-forming multiblock copolymers of the type B1AB2CB1 and AB2CB3. For certain compositions, these two multiblock copolymers self-assemble to form mesocrystalline phases composed of binary A and C spheres with the ZnSC and ReO3 structures, respectively. It is discovered that the addition of symmetric ABC triblocks or AB diblocks to the two multiblock copolymers leads to the formation of different binary crystalline phases including symmetric binary crystalline phase of NaCl type for the A/C-component symmetric blends and asymmetric Cu2O, SnI4, TiO2, and CaF2 phases for the A/C-component asymmetric blends. In particular, the binary mesocrystals of the Cu2O and SnI4 structures observed in the blends of the tail-symmetric B1AB2CB1 terpolymers and the AB diblock copolymers are new stable phases which have not been observed in the B1AB2CB3 terpolymers melts. The theoretical results demonstrate that blending of block copolymers with specially designed multiblock terpolymers could provide an efficient route to fabricate binary mesocrystals.

Self-assembling block copolymers provide access to the fabrication of various ordered phases. In particular, the ordered spherical phases can be used to engineer soft mesocrystals with domain size at the 5–100 nm scales. Simple block copolymers, such as diblock copolymers, form a limited number of mesocrystals. However multiblock copolymers are capable to form more complex mesocrystals. We demonstrate that designed B1AB2CB3 multiblock terpolymers, in which the A- and C-blocks form spherical domains and the packing of these spheres can be controlled by changing the lengths of the middle and terminal B-blocks, self-assemble into various binary mesocrystals with space group symmetries of a large number of binary ionic crystals, including NaCl, CsCl, ZnS, α-BN, AlB2, CaF2, TiO2, ReO3, Li3Bi, Nb3Sn(A15), and α-Al2O3. This approach can be generalized to other terpolymers as well as to tetrapolymers to obtain ternary mesocrystals. Our study provides a new concept of macromolecular metallurgy for producing crystal phases in a mesoscale and thus makes multiblock copolymers a robust platform for the engineering of functional materials.

The stability of various spherical phases formed in conformationally asymmetric AB diblock and architecture asymmetric ABm miktoarm block copolymers is investigated using self-consistent field theory. Both the conformational and architecture asymmetries are unified into a parameter of conformationally asymmetric degree, ε. We find that a complex spherical phase, the σ phase, becomes stable and its phase region expands between bcc and hexagonal phases as increasing ε. Only for large conformational asymmetry, for example, ε = 9 (or m = 3), the A15 phase becomes stable in the region between the σ phase and the hexagonal phase and its phase region terminates at the intermediate segregation region. Compared with the σ phase, the A15 phase has more favorable interfacial energy by enabling the formation of larger spherical domains, and therefore, it becomes more stable in the region of more symmetric volume fraction and stronger segregation.

The self-assembly of block copolymer–homopolymer blends in bulk, as well as under the direction of periodic patterned surfaces, has been investigated by computer simulations of the time-dependent Ginzburg–Landau theory. Specifically, a small amount of homopolymers are added to regulate the spontaneous nucleation rate and substrate patterns are designed to control the position and orientation of the induced nuclei. The mechanism, validity and efficiency of this scheme is examined using 2D and 3D computer simulations of cylinder-forming block copolymer–homopolymer blends, demonstrating that large-scale perfectly ordered patterns can be produced by controlling the position and orientation of induced multiple nucleation events. This scheme, combining the nucleation event of block copolymer self-assembly with the direction of the patterned surface, can be used in the lithography technique of block copolymers to significantly improve the directing efficiency, i.e., the density multiplication.

The phase behaviors of diblock copolymers confined in thin films with two identical preferential surfaces are investigated using the self-consistent field theory. Around 20 morphologies, including centrosymmetric and non-centrosymmetric ones, are considered to construct the two-dimensional phase diagram with respect to the volume fraction and the film thickness, while the interaction parameter χN and the surface preferences are fixed. When these morphologies are classified into four categories of ordered phases—sphere, cylinder, perforated lamella (corresponding to gyroid phase in bulk), and lamella—the phase diagram directly reveals the impact of the film confinement on the order–order transitions as a function of volume fraction via the comparisons to those in bulk. Our results also provide a comprehensive understanding over the dependence of the structure formations on the film thickness for each volume fraction.

We used three-dimensional self-consistent field theory to investigate the micellization behavior of A-b-(B-alt-C)n multiblock terpolymers in the presence of a solvent that is selective to the terminal A-block. In particular, we focused on the effects of the incompatibility parameter between B and C, χBC, and the composition of the solvophilic A-block, fA, on the formation of micelles from ABC triblock and A(BC)3 multiblock terpolymers, respectively. We observed a general trend that a segmented packing of B- and C-layers along the axial direction of the micelles is favored than the coaxial packing with the increasing of χBC or decreasing of fA. The separation of B and C blocks within a micelle leads to the formation of a variety of multicompartment micelle morphologies, such as core–shell–corona spherical micelles, hamburgers, and bump-surface micelles, in the ABC triblock copolymers. In the A(BC)3 multiblock terpolymers, we discovered more fascinating micelles by implementing the SCFT simulation than by the DPD simulation. Besides the BC-segmented worm-like micelles, which have been found in the DPD simulation work, concentric multilayer spheres and vesicles can be formed by the solvent-induced effect when the solvophilic A-block is a majority component. The SCFT method provides an efficient way to screen promising molecular architectures for the ability to self-assemble into technologically promising hierarchical structures.

The self-assembling behavior of ABC linear triblock copolymer melts is systematically studied using the self-consistent field theory, focusing on the emergence and stability of the knitting-pattern (KP) phase. The KP is one of the most intriguing unconventional phases formed from ”frustrated” linear triblock copolymers, where the interaction between the two end blocks is much weaker than those between neighboring blocks. Specifically phase diagrams for linear ABC triblock copolymer melts are constructed by comparing the free energy of about 10 candidate structures, including the knitting patterns, three-color lamellae (L3), core–shell cylinders (CSC), perforated lamellae (PL), cylinders-within-lamellae (LC), triple/quadruple cylinders-on-cylinders (C3/C4), double/triple helices-on-cylinders (H2C/H3C), and perforated circular lamella-on-cylinders (PC). The results of the phase behavior are presented for three cases with increasing complexity of the block copolymers. First of all, we investigate the stable region of the KP phase in triblock copolymers with a uniform segment size. Second, we study the impact of the conformational parameters as well as the interaction asymmetry between neighboring blocks on the stability of the KP phase. Finally, we examine the stability region of the KP phase surrounded by LC, PL, L3, CSC, PC, and C4 phases for a model system with a specific set of parameters corresponding to those of the polystyrene–poly(ethylene-co-butylene)–poly(methyl methacrylate) (PS–PEB–PMMMA) samples.

The emergence and stability of superstructured cylindrical phases in frustrated ABC linear triblock copolymers are investigated by the self-consistent field theory. Our results reveal that the complex single/double/triple helices-on-cylinder phases are formed when the straight cylinders-on-cylinder and rings-on-cylinder phases are frustrated due to packing constraints. A free energy comparison indicates that the double and triple helical phases are stable, whereas the other cylinders-on-cylinder phases are metastable. In addition, the chirality and rotation for the helical supercylinders are considered in our calculations. Our theoretical prediction is consistent with a number of experimental observations.

The phase behaviors of A(BC)nBA′ linear multiblock terpolymers are investigated using the pseudo-spectral method of self-consistent field theory by varying the volume fractions of different blocks. The relative stability among the lamellae-in-lamellae structures with different BC internal layers is tuned by the volume fraction of the two long tails. A larger A volume fraction favors the formation of structures with fewer BC thin layers. When the volume fraction of A is increased further, a hierarchical cylinder phase can be formed because of the effect of the spontaneous curvature and vice versa. The separation between B and C significantly reduces the phase regime of the cylinder, especially for the case of small A volume fraction.

The ordering dynamics of directed self-assembly of cylinder-forming diblock copolymers is studied by cell dynamics simulations. The directing field, mimicking chemically or topologically patterned substrates, is in the form of hexagonally arranged potential wells attractive to minority blocks. Time evolution of the defect concentration is used to characterize the ordering dynamics of the self-assembled cylindrical structures of the block copolymers. When the period of the external potential, Ls, is a small integer multiple of the cylinder-to-cylinder distance, L0, of the block copolymer microphase, the defect concentration decays exponentially. The defect annihilation becomes slower as Ls is increased, and eventually, the exponential decay law is broken. When the ratio Ls/L0 is a square root of an integer, large polycrystalline grains with different orientations are observed. The results are consistent with available experimental and theoretical results.

The phase behaviors of multiblock terpolymer A(BC)nB (or A(BC)n) with equal volume fractions of A and compositional symmetric (BC)nB (or (BC)n) are investigated by using the pseudospectral method of the self-consistent mean field theory. These terpolymers can self-assemble into hierarchical lamellar phases of perpendicular or parallel lamellae within lamellae, and the number of B/C thin layers in the parallel phase can be varied. The relative stability among these hierarchical lamellar phases can be tuned by the three interaction parameters of χABN, χACN, and χBCN. Two-dimensional phase diagrams, the cross sections of the three-dimensional phase diagram, are determined in our calculations. Our conclusion that the perpendicular phase is stable only in the case of χACN ≪ χABN < χBCN is consistent with experimental observations by Bates’s group. In addition, our results suggest that the existence of the perpendicular phase is generic in both types of terpolymers: A(BC)nB and A(BC)n, with different values of n, even for the special case of A(BC)n, that is, an ABC linear terpolymer.

The phase behavior of A(BC)nBA multiblock copolymer melts is investigated using self-consistent mean-field theory (SCMFT). Solutions of the SCMFT equations corresponding to hierarchical lamellar structures are obtained. The free energy of these structures are used to construct phase diagrams. It is predicted that hierarchical lamellar structures with different number of “internal” BC layers can be formed. The number of BC layers of the stable lamellar structure is determined by the competition between chain entropy and interfacial energy. It is found that more BC layers are preferred when the interactions between A and BC blocks are much stronger than that between B and C blocks.

Self-assembly of ABC star triblock copolymers confined in cylindrical nanopores is studied using real-space self-consistent mean-field theory. Specifically, the investigation focuses on the confined self-assembly of a triblock copolymer which forms hierarchical lamellae in the bulk. Generically, the hierarchical lamellae can be parallel or perpendicular to the pore surfaces. Concentric rings of A and B/C lamellae are formed in the parallel case. The B/C layers further form B/C domains. The number of B/C domains is controlled by the pore size. In the perpendicular case, the B/C layers are arranged alternatively along the pore axis. The stability of these observed structures is analyzed.

The phase behaviors of A(BC)nBA′ linear multiblock terpolymers are investigated using the pseudo-spectral method of self-consistent field theory by varying the volume fractions of different blocks. The relative stability among the lamellae-in-lamellae structures with different BC internal layers is tuned by the volume fraction of the two long tails. A larger A volume fraction favors the formation of structures with fewer BC thin layers. When the volume fraction of A is increased further, a hierarchical cylinder phase can be formed because of the effect of the spontaneous curvature and vice versa. The separation between B and C significantly reduces the phase regime of the cylinder, especially for the case of small A volume fraction.