Cylinder-gyroid-lamella transitions in diblock copolymer solutions: A simulated annealing study

Self-assembly of colloids with competing interactions confined in spheres

At low temperatures, colloidal particles with short-range attractive and long-range repulsive interactions can form various periodic microphases in bulk. In this paper, we investigate the self-assembly behaviour of colloids with competing interactions under spherical confinement by conducting molecular dynamics simulations. We find that the cluster, mixture, cylindrical, perforated lamellar and lamellar structures can be obtained, but the details of the ordered structures are different from those in bulk systems. Interestingly, the system tends to form more perforated structures when confined in smaller spheres. The mechanism behind this phenomenon is driven by the relationship between the energy of the ordered structures and the bending of the confinement wall, which is different from the mechanism in copolymer systems.

Phase behavior of ABC cyclic terpolymer melts: a simulation study

The phase behavior of ABC cyclic terpolymer melts is investigated using a simulated annealing technique. A ternary phase diagram is constructed by tuning the volume fractions of the three blocks (f_A, f_B, and f_C) in the case of symmetric interactions. 11 phases are predicted, including lamellae with spheres at the interfaces, lamellae with spheres inside a domain, lamellae with spheres inside domains, cylinders in perforated lamellae, [6.6.6] tiling patterns, lamella + cylinder, hierarchical double-gyroid, columnar piled disk, patched spheres, cylinders with spheres at the interfaces and double gyroid with spheres at the interfaces. In these structures, the end segments of the three blocks tend to distribute uniformly on the A/B, B/C, or A/C interfaces, which may result in superior mechanical properties of the structures in cyclic terpolymer systems than those of the same structures formed in star or linear terpolymer systems. The physical reason for the similarities and differences between the phases formed in ABC cyclic and star terpolymer systems is investigated. Our simulation results are compared with related experimental observations and theoretical calculations.

Formation Mechanisms of Porous Particles from Self-Assembly of Amphiphilic Diblock Copolymers inside an Oil-in-Water Emulsion Droplet upon Solvent Evaporation

The formation mechanisms of porous particles from self-assembly of amphiphilic diblock copolymers inside an oil-in-water emulsion droplet upon evaporation of the organic solvent are investigated based on Monte Carlo simulations for the first time. A morphological diagram of particles is constructed as a function of the surfactant concentration (φ) and the copolymer composition characterized by the volume fraction of the hydrophilic B block ( f_B). Particles with various morphologies are predicted. Morphological sequences from non-porosity to closed-porosity to capsules and finally to open-porosity particles are usually observed with increasing φ when f_B ≤ 1/2, with the only exception that capsules do not occur when f_B = 1/6. Furthermore, the critical φ value for a given morphological transition usually decreases with increasing f_B. Micelles are always observed at higher φ regions when f_B > 1/2. It is found that the specific surface area falls on almost the same regime for particles with the same kind of morphology, indicating that the morphology of a particle largely determines its specific surface area. The chain stretching varies with the particle morphology. It is the presence of the surfactant that makes the formation of porous particles possible, while when φ > 0, multiple morphological transitions can be induced by changing f_B. In the process of organic solvent removal, the value of f_B may affect the shape of pores inside the droplet and hence leads to the f_B dependence of the morphological sequences. When the solvent evaporation is not too fast, the resulting morphological sequence does not depend on the evaporation rate. Our results are compared with related experiments.

Effect of Chain Architecture on Self-Assembled Aggregates from Cyclic AB Diblock and Linear ABA Triblock Copolymers in Solution

The self-assembly behaviors of two block copolymers with the same chain length but different chain architectures (cyclic AB, linear ABA) in B-selective solvents are investigated using Monte Carlo simulations. A morphological transition sequence, from spherical micelles to cylindrical micelles, to vesicles and then to multicompartment vesicles, is observed for both copolymer systems when the interaction between the solvophobic A-block and the solvent is increased. In particular, toroidal micelles could be formed in triblock systems due to the presence of the bridging chains at the parameter region between cylindrical micelles and vesicles whereas disklike micelles are formed in cyclic systems. The simulation results demonstrated that the architecture of block copolymers could be used to regulate the structural characteristics and thermal stability of these self-assembled aggregates.

Molecular dynamics simulation of thermotropic bolaamphiphiles with a swallow-tail lateral chain: formation of cubic network phases

T-shaped bolaamphiphiles (TBA) with a swallow-tail lateral chain have been found to provide a fertile platform to produce complex liquid crystalline phases that are accessible through changes of temperature and lateral chain length and design. In this work, we use molecular simulations of a simple coarse-grained model to map out the phase behavior of this type of molecules. This model is based on the premise that the crucial details of the fluid structure stem from close range repulsions and the strong directional forces typical of hydrogen bonds. Our simulations confirm that TBAs exhibit a rich phase behavior upon increasing the length of their lateral chain. The simulations detect a double gyroid phase and an axial-bundle columnar phase which bear some structural resemblance to those found in the experiment. In addition, simulations predict two cocontinuous phases with 3D-periodicity: the "single" diamond and the "single" plumber's nightmare phase. Our analysis of energetic and entropic contributions to the free energy of phases formed by TBA with either swallow-tail or linear side-chains suggest that the 3D-periodic network phases formed by the former are stabilized by the large conformation entropy of the side-chains.

Phase behavior of ABC-type triple-hydrophilic block copolymers in aqueous solutions

The phase behavior of symmetric ABC triple-hydrophilic triblock copolymers in concentrated aqueous solutions is investigated using a simulated annealing technique. Two typical cases, in which the hydrophilicity of the middle B-block is either stronger or weaker than that of the end A- and C-blocks, are studied. In these two cases, a variety of phase diagrams are constructed as a function of the volume fraction of the B-block and the copolymer concentration ([Formula: see text] for both non-frustrated and frustrated copolymers. Structures, such as two-color alternatingly packed cylinders or gyroid, and lamellae-in-lamellae etc. that do not occur in the melt system, are obtained in solutions. Rich phase transition sequences, especially re-entrant phase transitions involving complex continuous networks of alternating gyroid and alternating diamond are observed for a given copolymer with decreasing [Formula: see text] . The difference in hydrophilicity among different blocks can result in inhomogeneous distribution of solvent molecules in the morphology, and with the decrease of [Formula: see text] , the distribution of solvent molecules presents a non-monotonic variation. This results in a non-monotonic variation of the effective volume fraction of each domain with the decrease of [Formula: see text] , which induces the re-entrant phase transitions. The presence of a good solvent for all the blocks can cause changes in the effective segregation strengths between different blocks and also in chain conformations, hence can alter the bulk phases and results in the occurrence of new structures and phase transitions. Especially, structures having A-C interfaces or A-C mixed domains can be obtained even in the non-frustrated copolymer systems, and structures obtained in the frustrated systems may be similar to those obtained in the non-frustrated systems. The window of the alternating gyroid structures may occupy a large part of the phase diagram for non-frustrated copolymers with stronger B-hydrophilicity. This behavior can be used to tune the self-assembled structures of block copolymers.

Engineering Gyroid-Structured Functional Materials via Templates Discovered in Nature and in the Lab

In search of optimal structures for functional materials fabrication, the gyroid (G) structure has emerged as a promising subject of widespread research due to its distinct symmetry, 3D interconnected networks, and inherent chiral helices. In the past two decades, researchers have made great progress fabricating G-structured functional materials (GSFMs) based on G templates discovered both in nature and in the lab. The GSFMs demonstrate extraordinary resonance when interacting with light and matter. The superior properties of GSFMs can be divided into two categories based on the dominant structural properties, namely, dramatic optical performances dominated by short-range symmetry and well-defined texture, and effective matter transport due to long-range 3D interconnections and high integrity. In this review, G templates suitable for fabrication of GSFMs are summarized and classified. State-of-the-art optical applications of GSFMs, including photonic bandgap materials, chiral devices, plasmonic materials, and matamaterials, are systematically discussed. Applications of GSFMs involved in effective electron transport and mass transport, including electronic devices, ultrafiltration, and catalysis, are highlighted. Existing challenges that may hinder the final application of GSFMS together with possible solutions are also presented.

Single Step Process for Self-Assembled Block Copolymer Patterns via in Situ Annealing during Spin-Casting

We demonstrated a simple and time-efficient processing method for facilitating a microphase separation of block copolymers (BCPs) based on a single step of spin-casting with low volatile solvent and in situ annealing. Well-ordered lamellar patterns of poly(styrene-b-methyl methacrylate) BCP films having wide range of molecular weights (51-235 kg/mol) were fabricated by a single 3 min process of spin-casting, even without the conventional pretreatment of substrate neutralization. The formation of this well-ordered lamellar structure is attributed to a synergetic effect between slow solvent evaporation and thermal energy that may provide an efficient cooling profile for the BCP film during the spin-casting process.

Direct Visualization of Order-Order Transitions in Silicon-Containing Block Copolymers by Electron Tomography

Here, we aim to comprehend the mechanism of the order-order transition (OOT) from nonequilibrium, metastable phase to equilibrium phase. Polystyrene-block-polydimethylsiloxane (PS-PDMS) block copolymer (BCP) bulks with metastable cylinder (C) and double gyroid (G) phases can be obtained from lamellae (L) forming PS-PDMS by simply tuning the selectivity of casting solvent. The recovery of the intrinsic L phase can be achieved by thermal annealing through OOT. Time-resolved small-angle X-ray scattering (SAXS) experiments are carried out to reveal the variation of the structural evolution in reciprocal space during annealing. The structural evolution in real space is directly visualized by using electron tomography (i.e., 3D transmission electron microscopy (TEM)). As a result, combining the time-resolved scattering experiments and the morphological observations from electron tomography offers new insights into the phase behaviors of the OOT of BCPs.

ORDERED MICROPHASES OF SYMMETRIC DIBLOCK COPOLYMER IN SELECTIVE SOLVENT: DISSIPATIVE PARTICLE DYNAMICS SIMULATION AND RADIAL DISTRIBUTION FUNCTION ANALYSIS

The self-assembly process of a symmetric diblock copolymer A5B5 in a selective solvent is studied using the dissipative particle dynamics method. Four types of morphologies can form, including lamellae, perforated lamellae, cylinders and spheres. Three phase diagrams are presented to demonstrate the effect of the concentration and interaction parameters on the ordered microstructures. Radial distribution function (RDF) characterizing the spatial correlations of the blocks is calculated. RDF not only confirms the morphological observations, but also provides more quantitative details. The evolution process of hexagonally packed cylinder and lamellae is also illustrated.

Formation of Two Kinds of Hexagonally Arranged Structures in ABC Triblock Copolymer Thin Films Induced by a Strongly Selective Solvent Vapor

An order-order transition (OOT) in the sequence of a hexagonally arranged core-shell cylinder to a double-hexagonally arranged dot in polystyrene-block-poly(butadiene)-block-poly(2-vinylpyridine) (SBV) triblock copolymer thin films is reported to be induced upon exposure to a solvent vapor that is strongly selective for the two end blocks. These two kinds of hexagonally arranged structures could form when the film thickness is 44, 123, and 223 nm. When the film thickness is decreased to 13 nm, the ordered structure is absent. The sizes of the core-shell cylinder structures formed with the same annealing time in films of different thickness are compared to address the effects of film thickness on the phase structure. The mechanism is analyzed from the total surface area of the blocks and the effective interaction parameter in the solvent vapor.

Early time kinetics of systems with spatial symmetry breaking

In this paper we present a study of the early stages of unstable state evolution of systems with spatial symmetry changes. In contrast to the early time linear theory of unstable evolution described by Cahn, Hilliard, and Cook, we develop a generalized theory that predicts two distinct stages of the early evolution for symmetry-breaking phase transitions. In the first stage the dynamics is dominated by symmetry-preserving evolution. In the second stage, which shares some characteristics with the Cahn-Hilliard-Cook theory, noise-driven fluctuations break the symmetry of the initial phase on a time scale which is large compared to the first stage for systems with long interaction ranges. To test the theory we present the results of numerical simulations of the initial evolution of a long-range antiferromagnetic Ising model quenched into an unstable region. We investigate two types of symmetry-breaking transitions in this system: disorder-to-order and order-to-order transitions. For the order-to-order case, the Fourier modes evolve as a linear combination of exponentially growing or decaying terms with different time scales.

Order-Order Transition of C → sdG → sL → S in ABC Triblock Copolymer Thin Film Induced by Solvent Vapor

The morphology transition of polystyrene-block-poly(butadiene)-block-poly(2-vinylpyridine) (SBV) triblock thin film induced in benzene vapor showing weak selectivity for PS is investigated. The order-order transitions (OOT) in the sequence of core-shell cylinders (C), sphere in 'diblock gyroid' (sdG), sphere in lamella (sL) and sphere (S) are observed. The projection along (111) direction in Gyroid phase (sdG(111)) is found to epitaxially grow from C(001) in the film. Instead of sdG(111), sdG(110)(0.1875) develops to the phase of sL. Consequently, the film experiences the transition sequence of sdG(111) → sdG(211) → sdG(110)(0.25)  → sdG(110)(0.1875) between C and sL. The mechanism is analyzed from the total surface area of the blocks.

Self-assembly of grafted Y-shaped ABC triblock copolymers in solutions

Self-assembled morphologies of grafted Y-shaped ABC triblock copolymers are investigated using a simulated annealing method. The block copolymers are composed of two incompatible arms (A and B) and a short stem (C), with the C-stems grafted onto a flat surface. A rich array of novel morphologies is discovered. The formation of these morphologies is controlled by polymer grafting density, the incompatibility between the A-B-blocks, as well as the quality and selectivity of the solvents. In particular, it is observed that solvent selectivity drives lateral and/or perpendicular microphase separation. A phase diagram for systems with low grafting density is constructed. It is predicted that multiple morphological transitions, such as these from mixed or core-shell micelles to internally segregated micelles, to hamburger-like micelles, to segmented wormlike micelles, to connected micelles, and to split micelles, can be induced by varying either the incompatibility between the two arms or the quality of the solvents. These results are consistent with previous experiments and theories.

Self-assembly of diblock copolymers confined in cylindrical nanopores

Self-assembly of AB diblock copolymers confined in cylindrical nanopores is studied using a simulated annealing technique. The pore diameter and surface preference are systematically varied to examine their effects on the self-assembled morphologies and the chain conformations. For bulk lamella-forming and cylinder-forming diblock copolymers, novel structures such as helices and concentric (perforated) lamellae spontaneously form when the copolymers are confined in cylindrical pores. The observed equilibrium morphologies are compared with that obtained from experiments, theory, and other simulations. A simple model is proposed for symmetric diblock copolymers, which gives a reasonable description of the layer thickness for the concentric lamellae. It is found that chains near the pore surfaces are compressed relative to the bulk chains, which can be attributed to the existence of the surfaces. The dependence of the chain conformation on the degree of confinement and strength of the surface preference are reasonably explained. The energetics is discussed qualitatively and used to account for the appearance of the complex phase behavior observed for certain intermediate conditions.

Self-assembled morphologies of diblock copolymers confined in nanochannels: Effects of confinement geometry

The self-assembly of diblock copolymers confined in channels of various shaped cross sections is studied using a simulated annealing technique with the "single-site bond fluctuation" model. In the bulk, the asymmetric diblock copolymers used in this study form hexagonally packed cylinders with period L0. The cross sections of the confining channels are of different shapes including regular triangles, rectangles, squares, regular hexagons, regular octagons, and ellipses. For a given geometry, the channel size (characterized by one or two lengths) is varied from very small to several times of L0. It is found that the geometry and size of the confining channels have a large effect on the structure and symmetry of the self-assembled morphologies. Multiple packed cylinders with the symmetry of the confining channels are the major morphologies for low-symmetry cross sections such as triangle, rectangle, and square. More complex structures such as helices or stacked toroids spontaneously form when the confining channels are shaped such as a regular hexagon, a regular octagon, or an ellipse. The domain spacing of the self-assembled structures can be altered by the shape and size of the confining channels. Our results are consistent with available experiments. These results indicate that the self-assembled structures of block copolymers can be manipulated by the shape of the confining channels.

Simulation of the gyroid phase in off-lattice models of pure diblock copolymer melts

Particle-based molecular simulations of pure diblock copolymer (DBC) systems were performed in continuum space via dissipative particle dynamics and Monte Carlo methods for a bead-spring chain model. This model consisted of chains of soft repulsive particles often used with dissipative particle dynamics. The gyroid phase was successfully simulated in DBC melts at selected conditions provided that the simulation box size was commensurate with the gyroid lattice spacing. Simulations were concentrated at conditions where the gyroid phase is expected to be stable which allowed us to outline approximate phase boundaries. When more than one phase was observed by varying simulation box size, thermodynamic stability was discerned by comparing the Helmholtz free energy of the competing phases. For this purpose, chemical potentials were efficiently simulated via an expanded ensemble that gradually inserts/deletes a target chain to/from the system. These simulations employed a novel combination of Bennett's [J. Comput. Phys. 22, 245 (1976)] acceptance-ratio method to estimate free-energy differences and a recently proposed method to get biasing weights that maximize the number of times that the target chain is regrown. The analysis of the gyroid nodes revealed clear evidence of packing frustration in the form of an (entropically) unfavorably overstretching of chains, a phenomenon that has been suggested to provide the structural basis for the limited region of stability of the gyroid phase in the DBC phase diagram. Finally, the G phase and nodal chain stretching were also found in simulations with a completely different DBC particle-based model.

Simulated annealing study of asymmetric diblock copolymer thin films

We report a simulated annealing study of the morphology of asymmetric diblock copolymer thin films confined between two homogeneous and identical surfaces. We have focused on copolymers that form a gyroidal morphology in the bulk. The morphological dependence of the confined films on the film thickness and the surface-polymer interaction has been systematically investigated. From the simulations it is found that much richer morphologies can form for the gyroid-forming asymmetric diblock copolymer thin films, in contrast to the lamella-forming symmetric and cylinder-forming asymmetric diblock copolymer films. Multiple morphological transitions induced by changing the film thickness and polymer-surface interactions are observed.

Simulated annealing study of gyroid formation in diblock copolymer solutions

Conditions for the formation of gyroid structures in diblock copolymer solutions are examined using a simulated annealing technique. The simulations were performed on diblock copolymer systems of A(NA)-b-B(NB) (with NA<NB) in solvents that are selective to the A blocks. It is shown that gyroid structures form in a narrow range of block copolymer concentrations between the hexagonally packed cylindrical and the lamellar phases and at an almost constant B-monomer concentration. It is also shown that the gyroid structure is especially sensitive to the B-solvent interaction (epsilonBS) and the length of the B block (NB). Phase diagrams for the diblock copolymer solutions are constructed. These predicted results are consistent with previous experimental observations. The three-dimensional isosurface contour plots of the simulated gyroid structure shows two interpenetrating strut networks. The projection along the [111] direction of the simulated gyroid structure and the spherically averaged structure factor are in good agreement with previous experimental results.