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Compositionally asymmetric diblock copolymers provide an attractive platform for understanding the emergence of tetragonally close-packed, Frank–Kasper phases in soft matter. Block-polymer phase behavior is governed by a straightforward competition between chain stretching and interfacial tension under the constraint of filling space at uniform density. Experiments have revealed that diblock copolymers with insufficient conformational asymmetry to form Frank–Kasper phases in the neat-melt state undergo an interconversion from body-centered cubic (bcc) close-packed micelles to a succession of Frank–Kasper phases (σ to C14 to C15) upon the addition of minority-block homopolymer in the dry-brush regime, accompanied by the expected transition from bcc to hexagonally packed cylinders in the wet-brush regime. Self-consistent field theory data presented here qualitatively reproduce the salient features of the experimental phase behavior. A particle-by-particle analysis of homopolymer partitioning furnishes a basis for understanding the symmetry breaking from the high-symmetry bcc phase to the lower-symmetry Frank–Kasper phases, wherein the reconfiguration of the system into polyhedra of increasing volume asymmetry delays the onset of macroscopic phase separation.

Ring polymer self-consistent field theory is used to calculate the critical temperatures and heat capacities of an ideal Bose gas for an order of magnitude more particles than previously reported. A λ -transition indicative of Bose-Einstein condensation is observed as expected. Using a known proof of spatial mode entanglement in Bose-Einstein condensates, a relationship between boson exchange and quantum entanglement is established. This is done without the use of state vectors, since ring polymer quantum theory uses instead a thermal degree of freedom, sometimes called the “imaginary time”, to map classical statistical mechanics onto non-relativistic quantum mechanics through the theorems of density functional theory. It is shown that quantum phenomena, such as Bose-Einstein condensation, boson exchange, entanglement and contextuality, can be visualized in terms of merging and separating ring polymer threads in thermal-space. A possible extension to fermions is mentioned.

Block copolymers comprising chemically different comblike or bottlebrush blocks can self-assemble in selective solvents giving rise to spherical or wormlike micelles or to polymersomes. The self-consistent field theoretical framework is used for predicting relation between the set of architectural parameters of the blocks (polymerization degrees of the main and side chains, density of grafting of the side chains to the backbone) and structural properties and morphology of the self-assembled aggregates. In particular, it is demonstrated that replacing linear blocks by architecturally symmetrical bottlebrush ones allows tuning the morphology of the self-assembled solution nanostructures.

The effect of colloidal nanoparticles on the phase changes of the amphiphilic AB linear diblock, A1A2B, and A2B heteroarm star copolymers confined between two polymer brush substrates was investigated by using a real-space self-consistent field theory. By changing the concentrations of nanoparticles and polymer brushes, the phase structure of the amphiphilic AB copolymer transforms from lamellar to core-shell hexagonal phase to cylinder phase. The pattern of A2B heteroarm star copolymer changes from core-shell hexagonal phases to lamellar phases and the layer decreases when increasing the density of the polymer brushes. The results showed that the phase behavior of the system is strongly influenced by the polymer brush architecture and the colloidal nanoparticle numbers. The colloidal nanoparticles and the soft confined surface of polymer brushes make amphiphilic AB copolymers easier to form ordered structures. The dispersion of the nanoparticles was also investigated in detail. The soft surfaces of polymer brushes and the conformation of the block copolymers work together to force the nanoparticles to disperse evenly. It will give helpful guidance for making some new functional materials by nano etching technology, nano photoresist, and nanoprinting.

We investigate the solution self-assembly of a mixture of positively charged homopolymers and AB diblock copolymers, in which the A blocks are negatively charged, and the B blocks are neutral. The electrostatic complexation between oppositely charged polymers drives the formation of many ordered phases. The microstructures and phase diagrams are calculated using self-consistent field theory (SCFT) based on an ion-pair model with an equilibrium constant K to characterize the strength of binding between positively and negatively charged monomers. The effects of the charge ratio, representing the ratio of charges from the homopolymer over all charges from polymers in the system, on the ordered structure are systematically studied, both for hydrophobic and hydrophilic A blocks. The charge ratio plays an important role in determining the phase boundaries in the phase diagram of salt concentration versus polymer concentration. We also provide information about the varying tendency of the domain spacing and core size of the spherical phase when the charge ratio is changed, and the results are in good agreement with experiments. These studies provide a deep understanding of the self-assembled microstructures of oppositely charged diblock copolymer-homopolymer systems.

We constructed phase diagrams of conformationally asymmetric diblock copolymer A-B melts using the polymer self-consistent field (SCF) calculations of both the dissipative particle dynamics chain (DPDC) model (i.e., compressible melts of discrete Gaussian chains with the DPD non-bonded potential) and the “standard” model (i.e., incompressible melts of continuous Gaussian chains with the Dirac δ-function non-bonded potential) in the χN-ε plane, where χN and ε characterize, respectively, the repulsion and conformational asymmetry between the A and B blocks, at the A-block volume fraction f = 0.2 and 0.3. Consistent with previous SCF calculations of the “standard” model, σ and A15 are the only stable Frank–Kasper (FK) phases among the five FK (i.e., σ, A15, C14, C15 and Z) phases considered. The stability of σ and A15 is due to their delicate balance between the energetic and entropic contributions to the Helmholtz free energy per chain of the system, which, within our parameter range, increases in the order of σ/A15, Z, and C14/C15. While in general the SCF phase diagrams of these two models are qualitatively consistent, A15 is not stable for the DPDC model at the copolymer chain length N = 10 and f = 0.3; any differences in the SCF phase diagrams are solely due to the differences between these two models.

The polymers can be either dynamically tethered to or permanently grafted to the nanoparticle to produce polymer-functionalized nanoparticles. The surface mobility of polymer ligands with one end anchored to the nanoparticle can affect the surface pattern, but the effect remains unclear. Here, we addressed the influence of lateral polymer mobility on surface patterns by performing self-consistent field theory calculations on a modeled polymer-functionalized nanoparticle consisting of immobile and mobile brushes. The results show that except for the radius of nanoparticles and grafting density, the fraction of mobile brushes substantially influences the surface patterning of polymer-functionalized nanoparticles, including striped patterns and patchy patterns with various patches. The number of patches on a nanoparticle increases as the fraction of mobile brushes decreases, favored by the entropy of immobile brushes. Critically, we found that broken symmetry usually occurs in patchy nanoparticles, associated with the balance of enthalpic and entropic effects. The present work provides a fundamental understanding of the dependence of surface patterning on lateral polymer mobility. The work could also guide the preparation of diversified nanopatterns, especially for the asymmetric patchy nanoparticles, enabling the fundamental investigation of the interaction between polymer-functionalized nanoparticles.

The quantum-classical isomorphism for self-consistent field theory, which allows quantum particles in space-time to be represented as classical one-dimensional threads embedded in a five dimensional thermal-space-time, is summarized and used to explain a selection of quantum phenomena. Introduced by Feynman, and used for modern quantum simulations, the isomorphism, when phrased in a field-theoretic way, has been shown to be the same as quantum density functional theory, the theorems of which guarantee equivalent predictions with non-relativistic quantum mechanics. If the Feynman dimension is considered to be real, there is a duality between classical threads in five dimensions and quantum particles in four dimensions. Using the 5D picture, intuitive explanations are given for quantum phenomena including the uncertainty principle, tunnelling, geometric phase, and interference effects. Advantages of the 5D picture are presented, which include fewer postulates, no measurement problem, and the need for only classical concepts in the higher dimensional space. Limitations of the approach such as the interpretation of entanglement and spin are discussed.

The phase behavior of ABCBA linear pentablock terpolymers is investigated by using the 3-dimensional self-consistent field theory. In this study, phase diagrams are constructed and used to discuss how the self-assembled morphologies are influenced by the compositions and the Flory–Huggins interaction parameters ( χ ) among the three components by decreasing symmetric χN value (segregation strength, where N is the total degree of polymerization) from 80 to 30. In the segregation regime of χN from 80 to 50, the microstructures formed by ABCBA linear pentablock terpolymers according to the compositions of the components are very similar. In particular, diverse complex network structures (e.g. diamond, hexagonally perforated lamellae, Fddd , and gyroid) and binary crystalline phases of cylinders and spheres can be observed. This is mainly due to the fact that the two free ends of the A block in the ABCBA linear copolymer allow the macromolecules to relieve packing frustrations. However, in the intermediate system with symmetric χN  < 50, the aggregation of each components becomes weaker so that behavior of the pentablock chains is similar to triblock and diblock chains. Accordingly, the diamond and hexagonally perforated lamellae tend to transfer to gyroid and Fddd observed in linear diblock and triblock cases. Moreover, by altering the composition ratio of A/C and the length of the B block, alternatively arranged A/C spheres resemble ionic and metallic crystals (e.g. NaCl, CsCl, Li 3 Bi, and Nb 3 Sn) and alternating A/C cylinders with coordination numbers of A/C (equal to 4/4, 6/3, and 4/2) can be still observed by decreasing symmetric χN to 40 in the intermediate regime.

The self-consistent field theory (SCFT) was employed to numerically study the interaction and interpenetration between two opposing weak polyelectrolyte (PE) brushes formed by grafting weak PE chains onto the surfaces of two long and parallel columns with rectangular-shaped cross-section immersed in a salty aqueous solution. The dependences of the brush heights and the average degree of ionization on various system parameters were also investigated. When the brush separation is relatively large compared with the unperturbed brush height, the degree of interpenetration between the two opposing PE brushes was found to increase with increasing grafting density and bulk degree of ionization. The degree of interpenetration also increases with the bulk salt concentration in the osmotic brush regime. Numerical results further revealed that, at a brush separation comparable to the unperturbed brush height, the degree of interpenetration does not increase further with increasing bulk degree of ionization, bulk salt concentration in the osmotic regime and grafting density. The saturation of the degree of interpenetration with these system parameters indicates that the grafted PE chains in the gap between the two columns retract and tilt in order to reduce the unfavorable electrostatic and steric repulsions between the two opposing PE brushes. Based on salt ion concentrations at the midpoint between the two opposing brushes, a quantitative criterion in terms of the unperturbed brush height and Debye screening length was established to determine the threshold value of the brush separation beyond which they are truly independent from each other.