Dynamics of phase separation in copolymer-homopolymer mixtures

Hybrid Time-Dependent Ginzburg–Landau Simulations of Block Copolymer Nanocomposites: Nanoparticle Anisotropy

Block copolymer melts are perfect candidates to template the position of colloidal nanoparticles in the nanoscale, on top of their well-known suitability for lithography applications. This is due to their ability to self-assemble into periodic ordered structures, in which nanoparticles can segregate depending on the polymer–particle interactions, size and shape. The resulting coassembled structure can be highly ordered as a combination of both the polymeric and colloidal properties. The time-dependent Ginzburg–Landau model for the block copolymer was combined with Brownian dynamics for nanoparticles, resulting in an efficient mesoscopic model to study the complex behaviour of block copolymer nanocomposites. This review covers recent developments of the time-dependent Ginzburg–Landau/Brownian dynamics scheme. This includes efforts to parallelise the numerical scheme and applications of the model. The validity of the model is studied by comparing simulation and experimental results for isotropic nanoparticles. Extensions to simulate nonspherical and inhomogeneous nanoparticles are discussed and simulation results are discussed. The time-dependent Ginzburg–Landau/Brownian dynamics scheme is shown to be a flexible method which can account for the relatively large system sizes required to study block copolymer nanocomposite systems, while being easily extensible to simulate nonspherical nanoparticles.

Solution landscapes of the diblock copolymer-homopolymer model under two-dimensional confinement

We investigate the solution landscapes of the confined diblock copolymer and homopolymer in two-dimensional domain by using the extended Ohta-Kawasaki model. The projection saddle dynamics method is developed to compute the saddle points with mass conservation and construct the solution landscape by coupling with downward and upward search algorithms. A variety of stationary solutions are identified and classified in the solution landscape, including Flower class, Mosaic class, Core-shell class, and Tai-chi class. The relationships between different stable states are shown by either transition pathways connected by index-1 saddle points or dynamical pathways connected by a high-index saddle point. The solution landscapes also demonstrate the symmetry-breaking phenomena, in which more solutions with high symmetry are found when the domain size increases.

Pathways connecting two opposed bilayers with a fusion pore: a molecularly-informed phase field approach

A phase field model with two phase fields, representing the concentration and the head-tail separation of amphiphilic molecules, respectively, has been constructed using an extension of the Ohta-Kawasaki model (Macromolecules, 1986, 19, 2621-2632). It is shown that this molecularly-informed phase field model is capable of producing various self-assembled amphiphilic aggregates, such as bilayers, vesicles and micelles. Furthermore, pathways connecting two opposed bilayers with a fusion pore are obtained by using a combination of the phase field model and the string method. Multiple fusion pathways, including a classical pathway and a leaky pathway, have been obtained depending on the initial separation of the two bilayers. The study shed light on the understanding of the membrane fusion pathways and, more importantly, laid a foundation for further investigation of more complex membrane morphologies and transitions.

Large scale three dimensional simulations of hybrid block copolymer/nanoparticle systems

Block copolymer melts self-assemble in the bulk into a variety of nanostructures, making them perfect candidates to template the position of nanoparticles. The morphological changes of block copolymers are studied in the presence of a considerable filling fraction of colloids. Furthermore, colloids can be found to assemble into ordered hexagonally close-packed structures in a defined number of layers when softly confined within the phase-separated block copolymer. A high concentration of interface-compatible nanoparticles leads to complex long-lived block copolymer morphologies depending on the polymeric composition. Macrophase separation between the colloids and the block copolymer can be induced if colloids are unsolvable within the matrix. This leads to the formation of ellipsoid-shaped polymer-rich domains elongated along the direction perpendicular to the interface between block copolymer domains.

Ashura Particles: Experimental and Theoretical Approaches for Creating Phase-Separated Structures of Ternary Blended Polymers in Three-Dimensionally Confined Spaces

Unique morphologies were found in binary and ternary polymer blended particles, including Ashura-type phase separation, which has three different polymer components on the particle surface. The morphologies of phase-separated structures in the binary polymer blended particles are discussed in terms of the surface tensions of the blended polymers. Structural control of ternary polymer blended particles was achieved based on the combination of polymers by examining binary polymer blended particles. A theoretical approach based on the Cahn-Hilliard equations gives identical morphologies with the experimental results. This work opens the way to creating polymer particles with sophisticated nanostructures by controlling their morphologies as predicted by theoretical simulations.

Defect Patterns from Controlled Heterogeneous Nucleations by Polygonal Confinements

Defects are often observed in crystalline structures. To regulate the formation or annihilation of defects presents an interesting question. In this work, we propose a method to fabricate defect patterns composed of regularly distributed steady "programmed defects", which is proceeded via the heterogeneous nucleation of a hexagonal pattern from a homogeneous state. The nucleation process occurring in a model system of AB-diblock/C-homopolymer blends under polygonal confinement is modeled by the time-dependent Ginzburg-Landau theory and is simulated by the cell dynamics simulations. Specifically, we demonstrate the validity of this method by means of three polygonal confinements including square, pentagon, and octagon, which have mismatched angles with the hexagonal lattice. Each corner or side of the polygons induces a nucleation event separately. Two nucleated domain grains by two neighboring corners or sides exhibit incommensurate orientations, and thus their merging leads to a radial line of clustered defects in the form of five-seven pairs. As a result, these radial lines constitute a radial pattern of defects, and their number is equal to the side number of the polygon. The distance of five-seven defect pairs is dictated by the incommensurate angle between two neighboring grains, which is similar to that of defects in hard crystals. This method can be extended to fabricate diverse defect patterns by programming the nucleation agents beyond simple polygonal confinements.

Transformation of Block Copolymer Nanoparticles from Ellipsoids with Striped Lamellae into Onionlike Spheres and Dynamical Control via Coupled Cahn-Hilliard Equations

Annealing of block copolymers has become a tool of great importance for the reconfiguration of nanoparticles. Here, we present the experimental results of annealing block copolymer nanoparticles and a theoretical model to describe the morphological transformation of ellipsoids with striped lamellae into onionlike spheres. A good correspondence between the experimental findings and predictions of the model was observed. The model based on finding the steepest direction of descent of an appropriate free energy leads to a set of Cahn-Hilliard equations that correctly describe the dynamical transformation of striped ellipsoids into onionlike spheres and reverse onionlike particles, regardless of the nature of the annealing process. This universality makes it possible to describe a variety of experimental conditions involving nanoparticles underlying a heating process. A notable advantage of the proposed approach is that it enables selective control of the interaction between the confined block copolymer and the surrounding medium. This feature endows the model with a great versatility to enable the reproduction of several combined effects of surfactants in diverse conditions, including cases with reverse affinities for the block copolymer segments. A phase diagram to describe a variety of morphologies is presented. We employ the relationship between the temperature-dependent Flory-Huggins parameter and the width of the interfaces to account for changes in temperature due to the heating process. Simulation results correctly show how the transformation evolves as the temperature increases. This increment in temperature corresponds to progressively smaller values of the interfacial width. We anticipate that the proposed approach will facilitate the design and more precise control of experiments involving various kinds of annealing processes.

Perfectly Ordered Patterns Formed by a Heterogeneous Nucleation Process of Block Copolymer Self-Assembly Under Rectangular Confinement

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.

Modeling size controlled nanoparticle precipitation with the co-solvency method by spinodal decomposition

The co-solvency method is a method for the size controlled preparation of nanoparticles like polymersomes, where a poor co-solvent is mixed into a homogeneous copolymer solution to trigger precipitation of the polymer. The size of the resulting particles is determined by the rate of co-solvent addition. We use the Cahn-Hilliard equation with a Flory-Huggins free energy model to describe the precipitation of a polymer under changing solvent quality by applying a time dependent Flory-Huggins interaction parameter. The analysis focuses on the characteristic size R of polymer aggregates that form during the initial spinodal decomposition stage, and especially on how R depends on the rate s of solvent quality change. Both numerical results and a perturbation analysis predict a power law dependence R∼s(-⅙), which is in agreement with power laws for the final particle sizes that have been reported from experiments and molecular dynamics simulations. Hence, our model results suggest that the nanoparticle size in size-controlled precipitation is essentially determined during the spinodal decomposition stage.

Frustrated phases under three-dimensional confinement simulated by a set of coupled Cahn-Hilliard equations

We numerically study a set of coupled Cahn-Hilliard equations as a means to find morphologies of diblock copolymers in three-dimensional spherical confinement. This approach allows us to find a variety of energy minimizers including rings, tennis balls, Janus balls and multipods among several others. Phase diagrams of confined morphologies are presented. We modify the size of the interface between microphases to control the number of holes in multipod morphologies. Comparison to experimental observation by transmission electron microtomography of multipods in polystyrene-polyisoprene diblock copolymers is also presented.

Regulating block copolymer phases via selective homopolymers

The phase behavior of strongly segregated AB diblock copolymer and selective C homopolymer blends is examined theoretically using a combination of strong stretching theory (SST) and self-consistent field theory (SCFT). The C-homopolymer is immiscible with the B-blocks but strongly attractive with the A-blocks. The effect of homopolymer content on the order-order phase transitions is analyzed. It is observed that, for AB diblock copolymers with majority A-blocks, the addition of the C-homopolymers results in lamellar to cylindrical to spherical phase transitions because of the A/C complexation. For diblock copolymers with minor A-blocks, adding C-homopolymers leads to transitions from spherical or cylindrical morphology with A-rich core to lamellae to inverted cylindrical and spherical morphologies with B-rich core. The results from analytical SST and numerical SCFT are in good agreement within most regions of the phase diagram. But the deviation becomes more obvious when the composition of A-blocks is too small and the content of added C-homopolymers is large enough, where the SCFT predicts a narrow co-existence region between different ordered phases. Furthermore, it is found that the phase behavior of the system is insensitive to the molecular weight of C-homopolymer.

Mesoscopic simulation of phase behaviors and structures in an amphiphile-solvent system

We have performed a three-dimensional simulation of mesoscopic structures in a mixture of AB amphiphilic molecule and C solvent by employing the density-functional theory under the conditions that (i) the size of the AB is much larger than C and (ii) the affinity between A and B is much larger than the affinity between B and C. First, we have calculated the free energy of five periodic structures, i.e., the lamellar phase, hexagonally packed cylinders, body-centered-cubic spheres, face-centered-cubic spheres, and gyroid phase for different sets of the concentration of AB (ϕ[over ¯]_{AB}) and the χ parameter (χ_{AC}). By comparing the free energies for these structures, the χ_{AC}-ϕ[over ¯]_{AB} phase diagram has been obtained. In addition to these periodic structures, it has been shown that nonperiodic structures such as spherical and rodlike micelles can be obtained although they might be metastable phase.

Domain patterns in a diblock copolymer-diblock copolymer mixture with oscillatory particles

We investigate the orientational order transition of striped patterns in microphase structures of diblock copolymer-diblock copolymer mixtures in the presence of periodic oscillatory particles. Under certain conditions, although the macrophase separation of a system is almost isotropic, microphase separation of one diblock copolymer takes place and becomes anisotropic gradually. By changing the oscillatory frequency and amplitude, the orientational order transition of a striped microphase structure from the state parallel to the oscillatory direction to the state perpendicular to the oscillatory direction is observed. We also find that the order transition occurs when we change the initial composition ratio. Furthermore, we examine the domain size and the orientational order parameter of microstructure in the process of orientational order transition. The results may provide guidance for experimentalists. This model system can also give a simple way to realize orientational order transition of soft materials by changing the oscillatory field.

Modeling self-assembly and phase behavior in complex mixtures

Using a variety of computational techniques, I investigate how the self-assembly of complex mixtures can be guided by surfaces or external stimuli to form spatially regular or temporally periodic patterns. Focusing on mixtures in confined geometries, I examine how thermodynamic and hydrodynamic effects can be exploited to create regular arrays of nanowires or monodisperse, particle-filled droplets. I also show that an applied light source and chemical reaction can be harnessed to create hierarchically ordered patterns in ternary, phase-separating mixtures. Finally, I consider the combined effects of confining walls and a chemical reaction to demonstrate that a swollen polymer gel can be driven to form dynamically periodic structures. In addition to illustrating the effectiveness of external factors in directing the self-organization of multicomponent mixtures, the selected examples illustrate how coarse-grained models can be used to capture both the equilibrium phase behavior and the dynamics of these complex systems.

Density functional simulation of spontaneous formation of vesicle in block copolymer solutions

The author carries out numerical simulations of vesicle formation based on the density functional theory for block copolymer solutions. It is shown by solving the time evolution equations for concentrations that a polymer vesicle is spontaneously formed from the homogeneous state. The vesicle formation mechanism obtained by this simulation agrees with the results of other simulations based on the particle models as well as experiments. By changing parameters such as the volume fraction of polymers or the Flory-Huggins interaction parameter between the hydrophobic subchains and solvents, the spherical micelles, cylindrical micelles, or bilayer structures can also be obtained. The author also shows that the morphological transition dynamics of the micellar structures can be reproduced by controlling the Flory-Huggins interaction parameter.

Dynamics of ternary mixtures with photosensitive chemical reactions: creating three-dimensionally ordered blends

Using computer simulations, we establish an approach for creating defect-free, periodically ordered polymeric materials. The system involves ABC ternary mixtures where the A and B components undergo a reversible photochemical reaction. In addition, all three components are mutually immiscible and undergo phase separation. Through the simulations, we model the effects of illuminating a three-dimensional (3D) sample with spatially and temporally dependent light irradiation. Experimentally, this situation can be achieved by utilizing both a uniform background light and a spatially localized, higher intensity light, and then rastering a higher-intensity light over the 3D sample. We first focus on the case where the higher-intensity light is held stationary and focused in a distinct region within the system. The C component is seen to displace the A and B within this region and replicate the pattern formed by the higher-intensity light. In effect, one can write a pattern of C onto the AB binary system by focusing the higher-intensity light in the desired arrangement. We isolate the conditions that are necessary for producing clearly written patterns of C (i.e., for obtaining sharp interfaces between the C and A/B domains). We next consider the effect of rastering a higher-intensity light over this sample and find that this light "combs out" defects in the AB blend as it moves through the system. The resulting material displays a defect-free structure that encompasses both a periodic ordering of the A and B domains and a well-defined motif of C. In this manner, one can create hierarchically patterned materials that exhibit periodicity over two distinct length scales. The approach is fully reversible, noninvasive, and points to a novel means of patterning with homopolymers, which normally do not self-assemble into periodic structures.

Exploiting photoinduced reactions in polymer blends to create hierarchically ordered, defect-free materials

Computer simulations reveal how photoinduced chemical reactions can be exploited to create long-range order in binary and ternary polymeric materials. The process is initiated by shining a spatially uniform light over a photosensitive AB binary blend, which thereby undergoes both a reversible chemical reaction and phase separation. We then introduce a well-collimated, higher intensity light source. Rastering this secondary light over the sample locally increases the reaction rate and causes formation of defect-free, spatially periodic structures. These binary structures resemble either the lamellar or hexagonal phases of microphase-separated diblock copolymers. We measure the regularity of the ordered structures as a function of the relative reaction rates for different values of the rastering speed and determine the optimal conditions for creating defect-free structures in the binary systems. We then add a nonreactive homopolymer C, which is immiscible with both A and B. We show that this component migrates to regions that are illuminated by the secondary, higher intensity light, allowing us to effectively write a pattern of C onto the AB film. Rastering over the ternary blend with this collimated light now leads to hierarchically ordered patterns of A, B, and C. The findings point to a facile, nonintrusive process for manufacturing high quality polymeric devices in a low-cost, efficient manner.

Harnessing light to create defect-free, hierarchically structured polymeric materials

Computer simulations reveal how photoinduced chemical reactions in polymeric mixtures can be exploited to create long-range order in materials with features that range from the submicron to the nanoscale. The process is initiated by shining a spatially uniform light on a photosensitive AB binary blend, which thereby undergoes both a reversible chemical reaction and a phase separation. When a well-collimated, higher intensity light is rastered over the sample, the system forms defect-free, spatially periodic structures. If a nonreactive homopolymer C is added to the system, this component localizes in regions that are irradiated with a higher intensity light, and one can effectively "write" a pattern of C onto the AB film. Rastering over the ternary blend with the collimated light now leads to hierarchically ordered patterns of A, B, and C. Because our approach involves homopolymers, it significantly expands the range of materials that can be fashioned into a periodic pattern. The findings point to a facile process for manufacturing high-quality polymeric components in an efficient manner.

Wetting-driven structure ordering of a copolymer/homopolymer/nanoparticle mixture in the presence of a modulated potential

We investigate pattern formation on a solid substrate of a diblock copolymer-homopolymer mixture containing doping wettable nanoparticles with a preferential attraction for one component of the copolymers, using a three-order-parameter model. The presence of doping nanoparticles under the surface-interaction modulation breaks the isotropy in the process of microphase-separation and macrophase-separation. This leads to the formation of orientational microphase and macrophase structures due to the interplay between the phase separation and wetting particle ordering under a modulated potential at the late stage. Simulations suggest that the microphase morphology and macrophase morphology can be changed through adjustment of the wetting strength, the amplitude as well as the period of the modulated potential. It provides some important insights for changing microphase and macrophase structures in polymer blends by wetting-driven spinodal decomposition.

Orientational order transition of the striped microphase structure of a copolymer-homopolymer mixture under oscillatory particles

Based on the three-order-parameter model, we investigate the orientational order transition of striped patterns in microphase structures of diblock copolymer-homopolymer mixtures in the presence of periodic oscillatory particles. Under suitable conditions, although the macrophase separation of a system is almost isotropic, the microphase separation of the system will be significantly perturbed by the oscillatory field, and composition fluctuations are suppressed anisotropically. The isotropy of the microphase will be broken up. By changing the oscillatory amplitude and frequency, we observe the orientational order transition of a striped microphase structure from the isotropic state to a state parallel to the oscillatory direction, and from the parallel state to a state perpendicular to the oscillatory direction. We examine, in detail, the microstructure and orientational order parameter as well as the domain size in the process of orientational order transition under the oscillatory field. We study also how the microphase structure changes with the composition ratio of homopolymers and copolymers in mixtures. The results suggest that our model system may provide a simple way to realize orientational order transition of soft materials.

Wetting-driven structure formation of a binary mixture in the presence of a mobile particle pinning potential

We investigate the pattern formation on the solid substrate of phase-separating films containing mobile wetting particles with a preferential attraction for one component of the mixtures. The presence of mobile particles under the surface-particle interaction modulation breaks the isotropy of the bulk phase-separating process, leading to the formation of orientational structure due to the interplay between phase separation and wetting particle ordering under a modulated pinning potential at the late stage. Simulations suggest that the phase-separation morphology can be changed through the adjustment of the wettable-phase-particle interaction and the surface-particle interaction. It provides some important insights into this "wetting-directed spinodal decomposition."