Real-space Self-consistent Field Theory Research Articles

The Effect of Colloidal Nanoparticles on Phase Separation of Block and Heteroarm Star Copolymers Confined between Polymer Brushes.

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.

Surface-induced phase transitions in thin films of dendrimer block copolymers

Abstract The phase morphologies and phase transitions of dendrimer block copolymer thin films confined between two homogeneous, planar hard substrates had been investigated by a three-dimensional real space self-consistent field theory (SCFT). From the perspectives of property and strength of the preferential substrate, when the film system confined within neutral substrates, the thinner film was easier to take the undulated and perpendicular cylinder phases. For the attractive preference of the substrate on block segment A, the polymer films tended to take the surface-wetting structures that was composed by block segment A. On the contrary, for the repulsive preference of the substrate on block segment A, a phase transition of cylinder-lamellae could be observed increasing with the relative surface strength of the preferential substrate.

Surface-Induced Nanostructures and Phase Diagrams of ABC Linear Triblock Copolymers under Spherical Confinement: A Self-Consistent Field Theory Simulation.

We investigate the nanostructures and phase diagrams of ABC linear triblock copolymers confined in spherical cavities by using real-space self-consistent field theory. Various 3D morphologies, such as spherical concentric lamellae, dumbbell-like cylinder, and rotational structures, are identified in the phase diagrams, which are constructed on the basis of the diameters of spherical cavities and the interaction between the polymers and preferential surfaces. We designate specific monomer-monomer interactions and block compositions, with which the polymers spontaneously form a cylindrical morphology in bulk, and firstly study morphology transformation with a neutral surface when a confining radius progressively increases. We then focus on phase morphologies under the preferential surfaces and consolidate them into phase diagrams. The spherical radius and the degree of preferential interactions can obviously induce the formation of a cylindrical morphology. Theoretical results correspond to an amount of recent experimental observations to a high degree and contribute to synthesising functional materials.

Self-assembly Behavior of Symmetrical Linear ABCA Tetrablock Copolymer: A Self-consistent Field Theory Study

ABCA tetrablock copolymers offer new opportunities for design of materials with novel structures. Using real-space self-consistent field theory and simulation, we systematically examined the self-assembly behavior of linear ABCA tetrablock copolymers in a 2D space. The simulation was carried out under conditions of symmetrical compositions and interactions. We focus on the influence of chain length ratio of block A and interactions between block A and other blocks B and C on the self-assembly behavior of the copolymer system. The simulation results show that most of the structures self-assembled by the ABCA tetrablock copolymers are centrosymmetric, such as diblock-like lamella phase, two kinds of lamellae with beads at interface, two kinds of hierarchical lamella phase, hexagonal honeycomb-like phase, lamella phase with mixed BC and hexagonal spheres with mixed BC. Furthermore, we find that a novel noncentrosymmetric Janus spheres can be obtained when the interaction between blocks B and C is strong, whereas a noncentrosymmetric lamella phase was obtained at weak interaction between blocks B and C. Phase diagrams for the ABCA tetrablock copolymers with different interaction strength between blocks B and C are constructed by comparing free energies of candidate ordered structures. In addition, studies on the metastable behavior of the system reveal that enthalpy plays an important role in the metastable behavior of the ABCA tetrablock copolymer system. Our work can provide useful guide for structure control of such kind of tetrablock copolymers in experiments.

Finite volume method for self-consistent field theory of polymers: Material conservation and application.

For the purpose of checking material conservation of various numerical algorithms used in the self-consistent-field theory (SCFT) of polymeric systems, we develop an algebraic method using matrix and bra-ket notation, which traces the Hermiticity of the product of the volume and evolution matrices. Algebraic tests for material conservation reveal that the popular pseudospectral method in the Cartesian grid conserves material perfectly, while the finite-volume method (FVM) is the proper tool when real-space SCFT with the Crank-Nicolson method is adopted in orthogonal coordinate systems. We also find that alternating direction implicit methods combined with the FVM exhibit small mass errors in the SCFT calculation. By introducing fractional cells in the FVM formulation, accurate SCFT calculations are performed for systems with irregular geometries and the results are consistent with previous experimental and theoretical works.

Surface-induced morphologies of ABC star triblock copolymer in spherical cavities

The morphologies and phase diagrams exhibited by symmetric ABC star triblock copolymer nanoparticles are investigated on the basis of real-space self-consistent field theory. The ABC star triblock copolymers were chosen to be tiling-forming with fixed polymer parameter and the spherical boundaries were modeled using the masking technique. We first study a number of examples where the ABC triblock copolymers confined in spherical cavities with neutral surface. Then, two types of spherical cavity distinct preferential surfaces are considered, including both A-block attractive and repulsive preferential surfaces. We aim at the effects due to various spherical cavity diameters and the degree of interactions between the polymer and the spherical surface. A variety of morphologies, such as ring-like structures, concentric sphere, and irregular cylinder, were identified in phase diagrams. The results show that both the degree of interactions and spherical diameters can influence the formation of morphologies so that ring-like structures and other novel structures could be obtained.

From vesicles to micelles: microphase separation of amphiphilic dendrimer copolymers in a selective solvent.

The microphase separation of amphiphilic dendrimer copolymers in a selective solvent with different excluded volume effects (αS) is investigated using three-dimensional real space self-consistent field theory. The morphological transition of disorder-to-order and order-to-order is observed by systematically regulating the excluded volume effect parameter, interaction parameter of block species, and the spacer length of the second generation of the dendrimer. The ordered segregates of the dendrimer solution are observed with a stronger excluded volume effect due to the strong depletion effect of solvent on the dendrimer. The relative magnitude between hydrophobic block B and hydrophilic block C is very important for microphase separation: when they are equal (NB = NC), a structural shift from vesicles to micelles has been found upon increasing the interaction parameter, and the region of disordered morphology is controlled by the interfacial free energy (Uint); when NB > NC, the vesicular morphologies overwhelmingly appear in the ordered region and then NC increases to close to NB, and the ordered aggregates take a shift from vesicles to micelles. Furthermore, the amphiphilic block C of the dendrimer is intended to enlarge to NC > NB, the micellar morphology is dominant in the ordered regime with a stronger excluded volume effect, which contributes to the decrease in the hydrophobic block repulsion that is affected by the decrease in the entropic free energy (-TS). The knowledge obtained from the microphase separation of dendrimer solution induced by the excluded volume effect of selective solvent is full of referential significance in understanding the morphological transition from vesicles to micelles for the amphiphile in the field of soft matter.

Side chain effect on the self-assembly of coil-comb copolymer by self-consistent field theory in two dimensions

The phase behavior of coil-comb copolymers A–(Bm+1Cm) (side chain number m = 1–5) is investigated by real-space self-consistent field theory (SCFT). Depending on the copolymer composition and architecture, eight two-dimensional ordered phases are observed, including two-color lamellae (LAM2), three-color lamellae (LAM3), hexagonal cylinders (HEX), core–shell hexagonal phase (CSH), hexagon outside hexagonal phase (HEX2), two interpenetrating tetragonal phase (TET2), lamellae with beads inside (LAM + BD), and lamellae with core–shell beads (LAM3 + CSB). When the volume fractions are comparable, i.e., fA ≈ fB ≈ fC, LAM_3 phase is found to be stable for m = 1 while the hexagonal phases (core–shell hexagonal phase CSH or hexagon outside hexagonal phase HEX2) are stable if m > 1. The phase region of the hexagonal phases HEX, CSH or HEX2 enlarges with increasing m. For short coil length, such as fA = 0.1, the phase diagram is complex, especially when m = 1. For longer coil length, the lamellae become the dominant phase. The phase transition from lamellar phase to hexagonal phase is observed with the increase of the side chain length when the side chain number m is large, which is in agreement with the experimental results. Our results give a good way to tailor the phase behavior of block copolymer and are very useful to further study the hierarchical structure of the coil-comb block copolymer.

Electric-field-induced sphere–cylinder phase transitions of diblock copolymers

Using the real space self-consistent field theory, the transitions between the body-centered-cubic phases and the cylindrical phases were studied for the diblock copolymers in the external electrostatic fields along three different orientations [001], [110], and [111]. The different orientations of the external fields resulted in the cylinder structures with various symmetries through the intermediate ellipsoid structures, where the cylinder-to-cylinder transition was also observed in the electric field along the orientation [110]. Furthermore, the size of ellipsoid was investigated by varying the strength of electric field, and the factors that affect the critical values of electric field strengths were also considered for the sphere–cylinder transitions in the electric fields with orientation [001]. Our findings are compared with the available experimental and theoretical works, which is in a general agreement.

Directed Self-Assembly of Block Copolymers on Sparsely Nanopatterned Substrates

Using real-space self-consistent field theory calculations (SCFT), directed self-assembly of block copolymers was explored on sparsely nanopatterned substrates. In this study, the lamellar structure forming block copolymers are confined between two surfaces. The top surface is strictly neutral, and the bottom one is periodically patterned with neutral interspacing (nanopattern), which forms a sparsely nanopatterned substrate. The dimension of nanopatterns is commensurate with periods of lamellar structure of the block copolymers self-assembling in bulk. By systematically varying the film thickness and interspacing between nanopatterns, the density multiplication of nanopattern in the film was investigated, and a variety of nanostructures not available in bulk (nonbulk structures) were observed.

Synthesis, characterization (molecular–morphological) and theoretical morphology predictions of linear triblock terpolymers containing poly(cyclohexadiene)

The synthesis via anionic polymerization of six linear triblock terpolymers with various sequences of blocks such as PS (polystyrene), PB [poly(butadiene)], PI [(poly(isoprene)] and PCHD [poly(1,3-cyclohexadiene)] is reported. The synthesis of the terpolymers was carried out by the use of anionic polymerization with high-vacuum techniques and sequential monomer addition. Molecular characterization of the samples was performed via size exclusion chromatography (SEC) and membrane osmometry (MO) to measure the polydispersity indices and the number-average molecular weights, respectively. Proton nuclear magnetic resonance spectroscopy (1H NMR) was employed to verify the microstructure type for the polydienes, as well as, to calculate the molar composition. Structural characterization of the terpolymers was performed via transmission electron microscopy (TEM) and small angle X-ray scattering (SAXS), and several morphologies were observed including one which has not been reported previously. Real-space self-consistent field theory (SCFT) without a priori knowledge about the symmetry of the periodic structures was used to elucidate the thermodynamics of the synthesized triblock copolymers.

Morphologies and phase diagrams of ABC star triblock copolymers confined in a spherical cavity

The morphologies and phase diagrams exhibited by symmetric ABC star triblock copolymer nanoparticles are investigated by using the real-space self-consistent field theory. A variety of three-dimensional morphologies, such as spherically concentric lamellae and ring-like structures, are identified in the triangular phase diagrams for the polymeric nanoparticles, depending on the volume fractions of the components and the interaction between the polymers and the confining spherical surface. We first study a number of examples where the confining radius and degree of interactions between the polymer and the spherical surface are fixed. A neutral surface that has no energetic preference on any ABC component induces ring-like structures in axisymmetry in an orbicular region of the phase diagram, which surrounds the central region where distorted cylinder phases are stable; a spherical surface that attracts one of the three polymer components induces distorted cylinders into the ring-like structures in a central region of the phase diagram; a spherical surface that repels one of the components stabilizes a lamella-like structure in spherical symmetry in a large region of the phase diagrams. We then focus on an example of polygonal tiling morphologies to study the effects due to variations of the spherical radius and the degree of interactions between the polymer and the spherical surface. The results show that the degree of interactions can obviously influence the formation of morphologies but the spherical radius only affects the polygonal tiling morphologies through slightly adjusting their natural arrangement in the neutral surface case.

Morphology diagrams for A2B copolymer melts: real-space self-consistent field theory

Morphology diagrams for A2B copolymer melts are constructed using real-space self-consistent field theory (SCFT). In particular, the effect of architectural asymmetry on the morphology diagram is studied. It is shown that asymmetry in the lengths of A arms in the A2B copolymer melts aids in the microphase separation. As a result, the disorder-order transition boundaries for the A2B copolymer melts are shown to shift downward in terms of χN, χ and N being the Flory's chi parameter and the total number of the Kuhn segments, respectively, in comparison with the A2B copolymers containing symmetric A arms. Furthermore, perforated lamellar (PL) and a micelle-like (M) microphase segregated morphologies are found to compete with the classical morphologies namely, lamellar, cylinders, spheres and gyroid. The PL morphology is found to be stable for A2B copolymers containing asymmetric A arms and M is found to be metastable for the parameter range explored in this work.

Self-assembly of linear triblock copolymers under cylindrical nanopore confinements

The self-assembly of linear ABC triblock copolymers under cylindrical confinements is investigated in two-dimensional space using the real-space self-consistent field theory. The effects of confinement degrees and preferential strengths on the triblock copolymer phase behaviors with special polymer parameters are first considered. On one hand, different confinement degrees cause different phase behaviors in nanopores with the neutral surfaces. Moreover, the strongly preferential surface fields can surpass the confinement degrees and volume fractions in determing the confined phase behaviors. On the other hand, in contrast, confined morphologies are more sensitive to the variations in the A-preferential surface field strength. Subsequently, the incompatibility degrees between different blocks are systematically varied under cylindrical nanopore confinements. Under cylindrical nanopore confinements, the morphologies are very sensitive to the variations in the incompatibility degrees. Meanwhile, nanopore confinements can affect order-disorder and order-order transition points in the bulk. The corresponding free, internal, and entropic energies as well as the order parameters are also quantificationally examined to deeply investigate the confined phase mechanisms, and a number of morphological transitions are confirmed to be of first-order. These findings may guide the design of novel nanostructures based on triblock copolymers by introducing confinements.

Phase diagrams of ABC linear triblock copolymers under nanopore confinements

The phase diagrams of triblock copolymers in cylindrical nanopores are investigated using the real-space self-consistent field theory in a two-dimensional space. We concentrate on pores with neutral surfaces and invariable diameters, whose rich variety of phases are considered to originate from pure geometric frustration. A series of triangular phase diagrams are constructed by varying the volume fractions for several sets of interaction parameters. These diagrams are sorted into three categories according to their interaction parameters. The confined phase diagrams exhibit several characteristic features that differ from those observed in the bulk. First, a rich variety of geometric frustration phases with strong symmetries, such as cylindrical and square, are observed in the triangular phase space because of the symmetry constraint in the geometric boundary. Second, the frustrated phases present some novel and complex features compared with those in the bulk, demonstrating that the confined morphologies much more sensitively depend on the subtle variation in the binary interaction parameters than those in the bulk. Meanwhile, the entropic energies of the symmetric melts with equal end block volume fractions are investigated to further understand the geometric frustration phase behaviors in the triangular phase diagrams. The reasonable formation mechanisms of the frustration phases are also discussed.

Nanostructures and phase diagrams of ABC star triblock copolymers in pore geometries

The nanostructures and phase diagrams of ABC star triblock copolymers in pore geometries are investigated using the real-space self-consistent field theory in two-dimensional space. Two types of pores with neutral surfaces, namely, pores with small and large diameters, are considered. A rich variety of nanostructures are exhibited by the ABC star triblock copolymers in these two types of pores, which differ from those observed in bulk and in other confinements. These structures include perpendicular undulating lamellae, concentric core-shell cylinders, polygonal tiling with cylindrical arrangements, and other complex structures. Triangular phase diagrams for the ABC star triblock copolymers are constructed. The small pores clearly affect the corner and central space of the phase diagrams by distorting the bulk structures into concentric arrangements. Meanwhile, the large pores induce the transformation of bulk structures into concentric structures in most of the phase space, but slightly affect the structures at the center of the phase diagrams. Furthermore, the order-order and order-disorder phase transitions, as well as the stable and metastable phases, in the triangular phase diagrams are examined by analyzing their free energies. These observations on the ABC star triblock copolymers in the pore geometries provide a deeper insight into the behavior of macromolecules in a confined system.

Microdomain morphology of cylinder-forming diblock copolymers under spherical shell confinement

The topic of self-assembly of cylinder-forming diblock copolymers (DBCPs) under spherical shell confinement in different surface fields is explored using real-space self-consistent field theory calculations (SCFT). Using this approach we observed various microstructures of cylinder-forming DBCPs at different confinement dimensions and surface fields. From detailed searching for the microdomain morphologies, an obvious conclusion is that the interactions between the confinement surface and the polymers have a great effect on the self-assembly. Most of the microstructures are unique and not reported in bulk or under planar and cylindrical confinements.

Hierarchical Nanostructures Self-Assembled from Diblock Copolymer/Homopolymer Blends with Supramolecular Interactions

Self-assembly of AB diblock copolymer/C homopolymer blends with reversible supramolecular interactions was studied by real-space self-consistent field theory. The reversible bond is formed between the B free end of the AB diblock copolymers and one end of the C homopolymers, and thereby the supramolecular blends consist of the AB diblock copolymers, C homopolymers, and supramolecular ABC terpolymers. The constitutions of the blends are dependent on the bonding strength and blend ratio. The change of the bonding strength and blend ratio leads to a series of hierarchically ordered alternating nanostructures. In these alternating nanostructures, the C homopolymers exhibit a swollen effect on the C substructures, and the coordination number of C cylinders decreases as the bonding strength increases. To gain the information about the hierarchical nanostructures in details, one-dimensional density profiles were plotted. The results were finally compared with the existing experimental findings, and an agreement was shown. The obtained results provided an insight into the role of the supramolecular interactions on the hierarchical nanostructure formations.

Surface-field-induced microstructures of asymmetric diblock copolymer nanoparticles

The microstructures of asymmetric diblock copolymer nanoparticles were investigated using real-space self-consistent field theory. The nanoparticle boundary provides a spherical confinement in which the highly deformable copolymers are forced into a closed and crowded space. The surface-field-induced effects on the microstructures were examined systematically, and a rich variety of novel structures, such as mixtures of distorted cylinders and spherical lamellae, was observed in the copolymer nanoparticles. Investigations of the free energies indicate that the surface-field-induced structural transitions are of the first order. The formation mechanism of the confinement-induced structures was reasonably elucidated, based on the interactions between the polymer chains and confinement boundaries, and on the spherical symmetry from confinements. Our theoretical results are in good agreement with available simulation and experimental observations. The asymmetric diblock copolymer nanoparticles show a rich variety of novel structures, such as mixtures of distorted cylinders and spherical lamellae, due to the surface-field-induced effects from the nanoparticle surfaces.

Geometric Frustration Phases of Diblock Copolymers in Nanoparticles

The geometric frustration phases are investigated for diblock copolymers in nanoparticles with neutral surfaces using real-space self-consistent field theory. First, a rich variety of geometric frustration phases with specific symmetries are observed in the polymer nanoparticles with invariable diameters by constructing the phase diagrams arranged as the volume fraction and Flory-Huggins interaction parameter. Most of the space in the phase diagram is filled with phases with strong symmetries, such as spherical or cubic symmetries, while a number of asymmetric or axisymmetric phases are located in a narrow space in the diagram. Then the geometric frustration phases are examined systematically for the diblock copolymers with special polymer parameters, and a rich variety of novel frustration phases with multilayered structures are observed by varying the diameters of the nanoparticles. Furthermore, the investigations on the free energies indicate that the transitions between these frustrated phases are first-order, and the formation mechanism of the frustration phases is reasonably elucidated.