The nucleation of quasicrystals remains a fundamental puzzle, primarily due to the absence of a periodic translational template. Here, we demonstrate that phasons - hidden degrees of freedom unique to quasiperiodic order - drive diverse nucleation pathways in icosahedral quasicrystals (IQCs). Combining a Landau free-energy model with the spring pair method, we compute distinct critical nuclei and their corresponding minimum energy paths. At low temperatures, a direct, symmetry-preserving pathway dominates. In contrast, higher temperatures promote a "symmetry detour" that reduces the nucleation barrier via a lower-symmetry critical nucleus. Remarkably, while the resulting bulk IQCs exhibit distinct real-space symmetries, they remain thermodynamically degenerate with identical diffraction patterns. We resolve this paradox within the high-dimensional projection framework, showing that phason shifts modulate real-space symmetry without altering bulk thermodynamics. Our findings establish phasons as the structural origin of pathway diversity, offering a new physical picture for the emergence of quasiperiodic order.

The Landau-Brazovskii model provides a theoretical framework for describing various phases arising from competing short- and long-range interactions in many physical systems. In this work, we investigate phase transitions among various ordered phases within the three-dimensional Landau-Brazovskii model. We construct the phase diagram of this model, which encompasses eight distinct phases, and systematically compute the transition pathways connecting various metastable and stable states using the Landau-Brazovskii saddle dynamics. Along each transition pathway, the critical nucleus is identified with some detailed analyses of its shape, energy barrier, and Hessian eigenvalues. Furthermore, we explore how the transition state is influenced by model parameters, revealing systematic trends in critical nucleus sizes and energy barrier heights. Our results provide a comprehensive characterization of the nucleation mechanisms within the Landau-Brazovskii model and offer valuable insights into the structural transformations of modulated-phase systems.

Discrete polymers with precise chemical structures and uniform chain lengths are utilized to quantitatively investigate the phase behaviors of binary blends composed of linear triblock copolymers and homopolymers. Phase diagrams and structural properties of four sets of blend systems (ABA/A, BAB/A, ABA/B, and BAB/B) were studied and compared. The emergence of a rich array of ordered mesostructures, including lamellae (LAM), hexagonally packed cylinders (HEX), body-centered cubic (BCC), hexagonally close-packed spheres (HCP), double gyroid (DG), and Frank-Kasper sigma and A15 phases, was shown to be regulated by systematically varying triblock architecture and the size and concentration of homopolymers. Self-consistent field theory (SCFT) calculations based on the freely jointed chain (FJC) model were performed to understand and explain the experimental observations. Both experimental and theoretical results suggest that homopolymers exhibit greater miscibility with triblock copolymers when the corresponding blocks are located at two ends (e.g., ABA/A and BAB/B) compared with the counterpart systems with homopolymer-miscible middle block (e.g., ABA/B and BAB/A). The distribution of the homopolymers depends critically on the relative chain length, where shorter chains disperse more uniformly within the domains. These findings underscore the pivotal role of copolymer architecture in governing the phase behavior of multicomponent systems.

Grain boundaries (GBs) are ubiquitous defects in crystalline materials. However, they remain less explored in block copolymer ordered phases. Here, we develop a self-consistent field theory framework to investigate GB structure and energetics in double-gyroid (DG) diblock copolymer networks. The GB energy landscape is obtained as a function of GB orientation, which reveals multiple local minima representing distinct network-switching GBs. Remarkably, the global minimum is a previously unidentified asymmetric-tilt network-switching GB (ATNS), exhibiting a lower energy than the experimentally observed (422) twin boundary (TB). Comparative analyses of representative low- (ATNS, (422) TB) and high-energy twist ((01̅1̅), (100) TNSs) GBs reveal that, unlike enthalpy-dominated hard matter, GB stability in DG networks is predominantly entropy-driven. Twist-type GBs generate new nodes and disrupt nodal coplanarity, causing chain packing frustration and large entropy penalties. Conversely, the ATNS preserves favorable network connectivity and minimizes conformational constraints on polymer chains, making it the energetically preferred GB.

Interparticle interactions with multiple length scales play a pivotal role in the formation and stability of quasicrystals. Choosing a minimal set of length scales to stabilize a given quasicrystal is a challenging problem. To address this challenge, we propose a symmetry-preserving screening method (SPSM) to design a Landau theory with a minimal number of length scales—referred to as the minimal Landau theory—that includes only the essential length scales necessary to stabilize quasicrystals. Based on a generalized multiple-length-scale Landau theory, SPSM first evaluates various spectral configurations of candidate structures under a hard constraint. It then identifies the configuration with the lowest free energy. Using this optimal configuration, SPSM calculates phase diagrams to explore the thermodynamic stability of desired quasicrystals. SPSM can design a minimal Landau theory capable of stabilizing the desired quasicrystals by incrementally increasing the number of length scales. Our application of SPSM has not only confirmed known behaviors in 10- and 12-fold quasicrystals but also led to a significant prediction that quasicrystals with 8-, 14-, 16-, and 18-fold symmetry could be stable within three-length-scale Landau models.

The influence of copolymer topology on the phase behavior of binary blends containing AB-type block copolymers and A homopolymers is studied by examining the instability modes of the system using the random-phase approximation. Depending on the concentration of the homopolymers, the blends could undergo microscopic or macroscopic phase separation, forming one ordered phase or two coexisting phases. The transition between these two distinct behaviors occurs at the Lifshitz point (LP), at which the wavelength of the most unstable mode diverges. We demonstrate that AB-type copolymers with identical block compositions but distinct chain topologies have different abilities to solubilize homopolymers, and their LPs exhibit two distinct, continuous (second-order) or discontinuous (first-order), features connected by a tricritical point. A more complex behavior featuring the coexistence of two instability modes with different finite length scales is also observed in multiblock copolymer/homopolymer blends. These behaviors are shown to be independent of specific chain models.

X-shaped liquid crystalline macromolecules (XLCMs) are obtained by tethering two flexible end A-blocks and two flexible side B-blocks to a semiflexible R-block. A rich array of ordered structures can be formed from XLCMs, driven by the competition between the interactions between the chemically distinct blocks and the molecular connectivity. Here, we report a theoretical study on the phase behavior of XLCMs with symmetric and asymmetric side blocks by using the self-consistent field theory (SCFT). A large number of ordered structures, including smectic phases, simple and giant polygons, are obtained as solutions of the SCFT equations. Phase diagrams of XLCMs as a function of the total length and asymmetric ratio of the side chains are constructed. For XLCMs with symmetric side blocks, the theoretically predicted phase transition sequence is in good agreement with experiments. For XLCMs with a fixed total side chain length, transitions between layered structure to polygonal phases, as well as between different polygonal phases, could be induced by varying the asymmetry of the side chains. The free energy density, domain size, side chain stretching, and molecular orientation are analyzed to elucidate mechanisms stabilizing the different ordered phases.

Two sets of binary blends were prepared by mixing core-selective B-homopolymers with either symmetric AB2 or asymmetric AB1(B2) linear-branched block copolymers, aiming to unravel the architectural effects on the formation of complex structures in polymer blends. The spatial distribution of homopolymers depends critically on both the copolymer architecture and homopolymer size. For short homopolymers, symmetric AB2 blends exhibit a cylinder-to-lamellae transition as the homopolymer loading increases, while their asymmetric counterparts follow a transition sequence from spheres to cylinders and then to double gyroids. When long homopolymers are added, symmetric AB2 block copolymers retain their initial cylindrical structures, whereas the Frank-Kasper sigma and C15 phases are stabilized in the asymmetric systems. These precisely defined blends eliminate molecular uncertainties arising from molecular weight distribution and compositional variations. Our results underscore the significant impact of architectural symmetry on homopolymer partitioning that leads to distinct equilibrium morphologies, providing key insights into rational structural engineering via simple blending.

The theoretical prediction that various binary macromolecular mesocrystals composed of A and C spheres could be formed by B1AB2CB3 pentablock terpolymers () offers a promising route to fabricate these intricately structured nanomaterials. However, experimental realization of this strategy has been impeded by the requirement of synthesizing precisely designed pentablock terpolymers. Here, we propose a conceptually new and technically simpler route to engineer binary macromolecular mesocrystals by using BA '/ABC/C ' B ternary block copolymer blends that are designed to replicate the phase behavior of B1AB2CB3 pentablock terpolymers. Using self-consistent field theory, we show that the ternary blends exhibit similar self-assembly behaviors as the pentablock copolymers, forming various mesocrystals with controllable coordination numbers. This study offers a simpler alternative to fabricating novel macromolecular mesocrystals and introduces a general design principle for emulating multiblock copolymers by block copolymer blends.

Block copolymers, obtained when two or more chemically distinct sub-chains or blocks are covalently linked together, tend to organize into assemblies of different shapes (spheres, cylinders, and lamellae) due to the competition between block–block repulsion and chain connectivity. The packing of these malleable assemblies leads to the formation of various ordered phases. For the case of spherical assemblies resembling soft spheres, their packing leads to, besides the commonly observed simple body-centred-cubic phase, the emergence of several complex spherical packing phases, i.e. the Frank–Kasper phases. Several mechanisms, including conformational asymmetry, copolymer architecture, and blending of different polymeric species, have been identified to stabilize the complex spherical packing phases. These developments shed light on our understanding of the packing of soft spheres self-assembled from soft matter.

We are pleased to announce the special topic of"AI for Polymers"published in the Chinese Journal of Polymer Sci-ence(CJPS).In recent years,the advancements in artificial intelligence(AI)techniques,including machine learning(particularly deep learning)and data-driven modeling,are reshaping how we design,synthesize and characterize polymers.

The fabrication of nanostructures from polycyclic aromatic hydrocarbons (PAHs) is highly attractive owing to their unique optical, electrical, and magnetic properties. However, the creation of uniform and well-defined PAH nanostructures by self-assembly still remains a significant challenge. Herein, we report that highly uniform hexagonal rods can be obtained from triphenylene (TP)-derived monomers by synchronized polymerization and self-assembly (SPSA). These rods have a single-domain columnar liquid crystalline structure in which columns formed from stacked TPs are along the long axis of the rods. The length/diameter ratios of the rods can be tuned over a wide range. Key factors to achieve SPSA of PAHs were analyzed, and the formation mechanism was clarified. In particular, it is observed that successful SPSA occurs below an upper critical temperature, which could be attributed to insufficient microphase separation between the side chains and the main chains and should be a general principle for SPSA. Furthermore, we demonstrate that the columnar stacking of TP units significantly promotes the intersystem crossing of the singlet excited state to the triplet excited state, resulting in simultaneous fluorescence and phosphorescence emission at room temperature. This work may be extended to a wide range of PAHs to regulate their self-assembly and light emission properties.

The creation of anisotropic nanoparticles (NPs) by polymerization and/or self-assembly (SA) has significantly promoted the applications of polymer nanomaterials in many fields. However, polymer nanorods are not easily accessible via conventional polymerization or SA. Here we report a one-step route to synthesize single-domain smectic liquid crystalline (LC) nanorods utilizing oriented attachment (OA) that was usually found in the synthesis of inorganic NPs, synchronized with polymerization and SA. The synchronization was achieved by developing a novel stabilizer derived from a thermo-responsive polyelectrolyte system. Mechanistic studies reveal that controlling the thermo-responsive behavior and the distribution of stabilizers on NPs enabled OA. The LC nanorods can further form hierarchical colloidal LCs, which show much larger light transmittance than that of non-LC nanorods. Moreover, we demonstrate that this LC system can be manipulated by an external magnetic field, thus providing a candidate material for magnetic-responsive display.

Quasicrystals are intriguing ordered structures characterized by the lack of translational symmetry and the existence of rotational symmetry. The tiling of different geometric units such as triangles and squares in two-dimensional space can result in a great variety of quasicrystals that could be realized by the self-assembly of liquid crystalline molecules. In this study, we introduce three self-similar dodecagonal tilings, including a novel Diamond-Square-Triangle pattern, composed of triangular and quadrangular tiles and examine their thermodynamic stability by using the self-consistent field theory applied to T-shaped liquid crystalline molecules. Specifically, we detail the inflation rules for the construction of these dodecagonal tilings and analyze their self-similarity, and show that these tilings can be viewed as projections of higher-dimensional periodic lattice points with projection windows. Using these dodecagonal tilings as initial configurations of the SCFT results in solutions corresponding to quasicrystals that could form from the T-shaped liquid crystalline molecules. The relative stability of these aperiodic phases is analyzed to obtain design rules that could stabilize quasicrystals. Meanwhile, we provide two criteria for distinguishing three dodecagonal quasicrystals and their approximants by analyzing their diffraction peaks. These findings shed new lighten on the discovery of new quasicrystals in soft materials.

The architecture of a block copolymer plays a pivotal role in tailoring its self-assembly behavior. In this work, discrete AB2 linear-branched block copolymer isomers with identical chemical structures but varying molecular symmetry were prepared and studied. Comparing to AB linear diblock copolymers, linear-branched AB2 counterparts exhibit significantly different assembly behaviors, which can be further regulated by adjusting the relative chain length of two B branches. A minor difference in the chain lengths of the B blocks results in an expansion in domain sizes and enhanced phase stability, while a larger asymmetry triggers phase transitions from a cylindrical structure to various spherical phases. The synergistic effects of the long and short B blocks effectively alleviate packing frustration, leading to a nonmonotonic deflection of the spherical/cylindrical phase boundary. The unique phase behaviors were substantiated by a self-consistent field theory study. This work demonstrates the feasibility and robustness of tailoring assembly behaviors by rational manipulation of chain architecture, providing insights into the underlying mechanism that stabilizes unconventional spherical phases.

Synthesizing anisotropic polymeric nanoparticles (NPs) with well-defined shapes, dimensions, and molecular orientations is a very challenging task. Herein, we report the synthesis of surprisingly highly uniform shape-anisotropic polymer NPs with uniaxial internal molecular orientation. Keys to our method are synchronized polymerization and self-assembly (SPSA), which can even be realized by regular dispersion polymerization. This is demonstrated using a monomer containing a rigid 4-nitroazobenzene (NAB) side group. The short nucleation period, the completion of microphase separation before molecular motion is frozen, and sufficient low particle/solvent interfacial tension are shown to be the origins of the highly uniform dimensions, single liquid crystal domains, and well-defined anisotropic shape of particles. The liquid crystallization ability of the polymers, control of molecular weight distribution, and the polymerization kinetics are identified as three key factors controlling the NP formation. The uniformity of these NPs facilitates their SA formation into colloidal crystals. The particles exhibit optically anisotropic properties depending on orientations and, in particular, show intriguing photoswitchable LC-glass (order-disorder) transition, which can be used for the detection of ultraviolet (UV) light and allows the fabrication of photoreversible colloidal films.

Interparticle interactions with multiple length scales play a pivotal role in the formation and stability of quasicrystals. Choosing a minimal set of length scales to stabilize a given quasicrystal is a challenging problem. To address this challenge, we propose an intelligent screening method (ISM) to design a Landau theory with a minimal number of length scales – referred to as the minimal Landau theory – that includes only the essential length scales necessary to stabilize quasicrystals. Based on a generalized multiple-length-scale Landau theory, ISM first evaluates various spectral configurations of candidate structures under a hard constraint. It then identifies the configuration with the lowest free energy. Using this optimal configuration, ISM calculates phase diagrams to explore the thermodynamic stability of desired quasicrystals. ISM can design a minimal Landau theory capable of stabilizing the desired quasicrystals by incrementally increasing the number of length scales. Our application of ISM has not only confirmed known behaviors in 10- and 12-fold quasicrystals but also led to a significant prediction that quasicrystals with 8-, 14-, 16-, and 18-fold symmetry could be stable within three-length-scale Landau models.

Extensive experimental studies have shown that numerous ordered phases can be formed via the self-assembly of T-shaped liquid crystalline molecules (TLCMs) composed of a rigid backbone, two flexible end chains and a flexible side chain. However, a comprehensive understanding of the stability and formation mechanisms of these intricately nanostructured phases remains incomplete. Here we fill this gap by carrying out a theoretical study of the phase behaviour of TLCMs. Specifically, we construct phase diagrams of TLCMs by computing the free energy of different ordered phases of the system. Our results reveal that the number of polygonal edges increases as the length of side chain or interaction strength increases, consistent with experimental observations. The theoretical study not only reproduces the experimentally observed phases and phase transition sequences, but also systematically analyzes the stability mechanism of the polygonal phases.

X-shaped liquid crystalline molecules (XLCMs) are obtained by tethering two flexible end A-blocks and two flexible side B-blocks to a rigid backbone (R). A rich array of ordered structures can be formed from XLCMs, driven by the competition between the interactions between the chemically distinct blocks and the molecular connectivity. Here, we report a theoretical study on the phase behaviour of XLCMs with symmetric and asymmetric side blocks by using the self-consistent field theory (SCFT). A large number of ordered structures, including stable smectic-A, triangle-square, pentagon and giant polygon, are obtained as solutions of the SCFT equations. Phase diagrams of XLCMs as a function of the total length and asymmetric ratio of the side chains are constructed. For XLCMs with symmetric side blocks, the theoretically predicted phase transition sequence is in good agreement with experiments. For XLCMs with a fixed total side chain length, transitions between layered structure to polygonal phases, as well as between different polygonal phases, could be induced by varying the asymmetry of the side chains. The free energy density, domain size, side-chain stretching , and molecular orientation are analyzed to elucidate mechanisms stabilizing the different ordered phases.

In this work, we study the nucleation of quasicrystals from liquid or periodic crystals by developing an efficient order-order phase transition algorithm, namely, the nullspace-preserving saddle search method. In particular, we focus on nucleation and phase transitions of the decagonal quasicrystal (DQC) based on the Lifshitz-Petrich model. We present the nucleation path of DQC from the liquid and demonstrate one- and two-stage transition paths between DQC and periodic crystals. We provide a perspective of the group-subgroup phase transition and nucleation rates to understand the nucleation and phase transition mechanisms involving DQC. These results reveal the one-step and multi-step modes of symmetry breaking or recovery in the phase transition from DQC, where the multi-step modes are more probable.