Thin films of block copolymer

Dynamic Programming for Chain Propagator Computation of Branched Block Copolymers in Polymer Field Theory Simulations

We present an algorithmic approach to optimize chain propagator computations in polymer field theory simulations, including self-consistent field theory (SCFT) calculations and field-theoretic simulations (FTSs). Propagator calculations for branched block copolymers often involve recursive structures and overlapping subproblems, resulting in redundant computations. By employing dynamic programming (DP) and encoding computational dependencies as strings, our method systematically eliminates these redundancies in mixtures of branched polymers. The algorithm achieves optimal time complexity for various polymeric systems, including star-shaped, comb, dendrimer polymers, and homopolymer mixtures, by reusing and aggregating propagators for symmetric and repetitive structures. This enhances computational efficiency and reduces memory usage, addressing a key limitation in developing versatile polymer field theory simulation software. Our approach streamlines the simulation of complex branched polymers without requiring manual software adjustments, facilitating more efficient workflows for polymer researchers. Furthermore, the method enables automated searches for inverse design by optimizing computations across diverse branched polymer architectures, contributing to the discovery and design of novel polymeric materials. The algorithm is implemented in open-source software, ensuring accessibility for further development and broader application in computational polymer science.

Equilibrium phase behavior of gyroid-forming diblock polymer thin films

Thin-film confinement of self-assembling block polymers results in materials with myriad potential applications-including membranes and optical devices-and provides design parameters for altering phase behavior that are not available in the bulk, namely, film thickness and preferential wetting. However, most research has been limited to lamella- and cylinder-forming polymers; three-dimensional phases, such as double gyroid (DG), have been observed in thin films, but their phase behavior under confinement is not yet well understood. We use self-consistent field theory to predict the equilibrium morphology of bulk-gyroid-forming AB diblock polymers under thin-film confinement. Phase diagrams reveal that the (211) orientation of DG, often observed in experiments, is stable between nonpreferential boundaries at thicknesses as small as 1.2 times the bulk DG lattice parameter. The (001) orientation is stable between modestly B-preferential boundaries, where B is the majority block, while a different (211)-oriented termination plane is stabilized by strongly B-preferential boundaries, neither of which has been observed experimentally. We then describe two particularly important phenomena for explaining the phase behavior of DG thin films at low film thicknesses. The first is "constructive interference," which arises when distortions due to the top and bottom boundaries overlap and is significant for certain DG orientations. The second is a symmetry-dependent, in-plane unit-cell distortion that arises because the distorted morphology near the boundary has a different preferred unit-cell size and shape than the bulk. These results provide a thermodynamic portrait of the phase behavior of DG thin films.

Sequential Self-Assembly of Polystyrene-block-Polydimethylsiloxane for 3D Nanopatterning via Solvent Annealing

This study aims to develop a strategy for the fabrication of multilayer nanopatterns through sequential self-assembly of lamella-forming polystyrene-block-polydimethylsiloxane (PS-b-PDMS) block copolymer (BCP) from solvent annealing. By simply tuning the solvent selectivity, a variety of self-assembled BCP thin-film morphologies, including hexagonal perforated lamellae (HPL), parallel cylinders, and spheres, can be obtained from single-composition PS-b-PDMS. By taking advantage of reactive ion etching (RIE), topographic SiO2 monoliths with well-ordered arrays of hexagonally packed holes, parallel lines, and hexagonally packed dots can be formed. Subsequently, hole-on-dot and line-on-hole hierarchical textures can be created through a layer-by-layer process with RIE treatment as evidenced experimentally and confirmed theoretically. The results demonstrated the feasibility of creating three-dimensional (3D) nanopatterning from the sequential self-assembly of single-composition PS-b-PDMS via solvent annealing, providing an appealing process for nano-MEMS manufacturing based on BCP lithography.

Directed Self-Assembly by Sparsely Prepatterned Substrates with Self-Responsive Polymer Brushes

The guiding pattern in the chemoepitaxially directed self-assembly (DSA) of block copolymers is often fabricated by periodically functionalizing homogeneously random copolymer brushes tethered on a substrate. The prepatterned copolymer brushes constitute a soft penetrable surface, and their two components can in principle locally segregate in response to the overlying self-assembly process of block copolymers. To reveal how the self-responsive behavior of the copolymer brushes affects the directing effect, we develop a dissipative particle dynamics model to explicitly include the prepatterned polymer brushes and implement it to simulate the DSA of a cylinder-forming diblock copolymer melt on the sparse pattern of polymer brushes. Through large-scale dynamic simulations, we identify the windows of the content of the random copolymer, the film thickness, and the diameter of the patterned spot, for the formation of perfectly ordered hexagonal patterns composed of perpendicular cylinders. Our dynamic simulations reveal that the random copolymer brushes grafted on the unpatterned area exhibit a remarkable self-responsive ability with respect to the self-assembly of the diblock copolymers overlying them, which may widen the effective window of the content of the random copolymer. Within the processing windows of these key parameters, defect-free patterns are successfully achieved both in simulations and in experiments with sizes as large as a few micrometers for 4-fold density multiplications. This work demonstrates that highly efficient computer simulations based on an effective model can provide helpful guidance for experiments to optimize the critical parameters and even may promote the application of DSA.

The mechanism underlying the transitions between stripes, helices, and stacked toroids in the cylindrical shell formed by AB diblock copolymers on a long nanocylinder

The self-assembly of block copolymers on nanocylinders has attracted a lot of interest due to its potential application in biomedicine and other fields. In this study, the self-assembly phase behavior of AB diblock copolymers on long nanocylinders in soft confinement has been studied by using a simulated annealing method. A square phase diagram of the morphology was constructed by increasing the number of chains of copolymers (cn) and the cylindrical diameter (D). As a result, morphological transitions from striped to helical and axially stacked toroids, as well as reversible transitions, started to appear. By analyzing the chain packing in a fan-shaped region and calculating the mean-square end-to-end distance (DEE2) of the copolymers and number of AB contacts, both types of transitions were found to be driven by the competition between conformational entropy and AB interfacial energy. The number of stripes increased and the helical angle decreased with the increase in cylinder diameter. The chirality of the helix was found to be random.

Boundary Frustration in Double-Gyroid Thin Films

Self-consistent field theory for thin films of AB diblock polymers in the double-gyroid phase reveals that in the absence of preferential wetting of monomer species at the film boundaries, films with the (211) plane oriented parallel to the boundaries are more stable than other orientations, consistent with experimental results. This preferred orientation is explained in the context of boundary frustration. Specifically, the angle of intersection between the A/B interface and the film boundary, the wetting angle, is thermodynamically restricted to a narrow range of values. Most termination planes in the double gyroid cannot accommodate this narrow range of wetting angles without significant local distortion relative to the bulk morphology; the (211)-oriented termination plane with the "double-wave" pattern produces relatively minimal distortion, making it the least frustrated boundary. The principle of boundary frustration provides a framework to understand the relative stability of termination planes for complex ordered block polymer phases confined between flat, nonpreferential boundaries.

Nonmonotonic polymer translocation kinetics through nanopores under changing surface-polymer interactions

Understanding the dynamics of polymers in confined environments is pivotal for diverse applications ranging from polymer upcycling to bioseparations. In this study, we develop an entropic barrier model using self-consistent field theory that considers the effect of attractive surface interactions, solvation, and confinement on polymer kinetics. In this model, we consider the translocation of a polymer from one cavity into a second cavity through a single-segment-width nanopore. We find that, for a polymer in a good solvent (i.e., excluded volume, u0 > 0), there is a nonmonotonic dependence of mean translocation time (τ) on surface interaction strength, ɛ. At low ɛ, excluded volume interactions lead to an energetic penalty and longer translocation times. As ɛ increases, the surface interactions counteract the energetic penalty imposed by excluded volume and the polymer translocates faster through the nanopore. However, as ɛ continues to increase, an adsorption transition occurs, which leads to significantly slower kinetics due to the penalty of desorption from the first cavity. The ɛ at which this adsorption transition occurs is a function of the excluded volume, with higher u0 leading to an adsorption transition at higher ɛ. Finally, we consider the effect of translocation across different size cavities. We find that the kinetics for translocation into a smaller cavity speeds up while translocation to a larger cavity slows down with increasing ɛ due to higher surface contact under stronger confinement.

Effect of Film Thickness on the Self-Assembly of CBABC Symmetric Pentablock Terpolymer Melts under 1D Confinement: A Dissipative Particle Dynamic Study

The study investigates the impact of film thickness on the phase behavior of pentablock terpolymers, denoted as C3B3A6B3C3, when subjected to wall confinement by utilizing the dissipative particle dynamics method. Phase diagrams were constructed to elucidate how factors such as block-block interaction strength, film thickness, and wall properties affect the self-assembly structures. In cases where the wall exhibits no preference for any of the blocks, lamellae phases with orientations perpendicular to the wall are observed. The order-disorder transition (ODT) temperature is found to be influenced by the interaction between the polymer and the wall in thin confinement scenarios. When the wall displays a preference for specific blocks, the orientation of lamellae structures undergoes variations. Lamellae tend to align parallel to the wall when the wall favors A or C blocks, and they orient perpendicularly when B blocks are favored. Furthermore, the mechanical properties of the lamellae structures are related to the conformations of the polymer chains. Structures where chains predominantly adopt a loop conformation exhibit enhanced elastic properties. The ratio of looping to bridging conformations can be adjusted by altering the film thickness and wall selectivity.

Distributions of Deuterated Polystyrene Chains in Perforated Layers of Blend Films of a Symmetric Polystyrene-block-poly(methyl methacrylate)

We have examined the spatial distributions of polymer chains in blend films of weakly segregated polystyrene-block-poly(methyl methacrylate) [P(S-b-MMA)] and deuterated polystyrene (dPS). By fine-tuning the composition (ϕ_PS+dPS = 63.8 vol %) of the total PS/dPS component and annealing temperature (230 and 270 °C), P(S-b-MMA)/dPS blend films mainly form perforated layers with a parallel orientation (hereafter PLs^//). The distributions of dPS in PLs^// were probed by grazing-incidence small-angle neutron scattering (GISANS) and time-of-flight neutron reflectivity (ToF-NR). GISANS and ToF-NR results offer evidence that dPS chains preferentially locate at the free surface and within the PS layers for blend films that were annealed at 230 °C. Upon annealing at 270 °C, dPS chains distribute within PS layers and perforated PMMA layers. Nevertheless, dPS chains still retain a surface preference for thin films. In contrast, such surface segregation of dPS chains is prohibited for thick films when annealed at 270 °C.

Self-consistent field theory study of polymer-mediated colloidal interactions in solution: Depletion effects and induced forces

Polymer-mediated colloidal interactions control the stability and phase properties of colloid-polymer mixtures that are critical for a wide range of important applications. In this work, we develop a versatile self-consistent field theory (SCFT) approach to study this type of interaction based on a continuum confined polymer solution model with explicit solvent and confining walls. The model is formulated in the grand canonical ensemble, and the potential of mean force for the polymer-mediated interaction is computed from grand potentials. We focus on the case of non-adsorbing linear polymers and present a systematic investigation on depletion effects using SCFT. The properties of confined polymer solutions are probed, and mean-field profiles of induced interactions are shown across different physical regimes. We expose a detailed parametric dependence of the interaction, concerning both attractive and repulsive parts, on polymer concentration, chain length, and solvent quality and explore the effect of wall surface roughness, demonstrating the versatility of the proposed approach. Our findings show good agreement with previous numerical studies and experiments, yet extend prior work to new regimes. Moreover, the mechanisms of depletion attraction and repulsion, along with the influence of individual control factors, are further discussed. We anticipate that this study will provide useful insights into depletion forces and can be readily extended to examine more complex colloid-polymer mixtures.

Phase behavior in thin films of weakly segregated block copolymer/homopolymer blends

We have demonstrated the phase behavior of substrate-supported films of a symmetric weakly segregated polystyrene-block-poly (methyl methacrylate), P(S-b-MMA), block copolymer and its blends with homopolymer polystyrene (PS) at different compositions. Upon increasing the content of added PS in the blends, lamellae (L), perforated layers (PL), double gyroid (DG) and cylinders (C) are obtained in sequence for films. Among these nanodomains, PL and DG only exist in a narrow ϕ_PS region (ϕ_PS denotes the volume fraction of PS). At ϕ_PS = 64%, tuning film thickness and annealing temperature can produce parallel PL or DG with {121}_DG lattice planes being parallel to the substrate surface. The effects of annealing temperature and film thickness on the formation of PL and DG are examined. In thin films with n ≈ 3 (n denotes the ratio of initial film thickness to inter-domain spacing), the PL phase solely exists regardless of temperature. However, for thick films with n ≈ 6 and 10, thermal annealing at the most accessible temperature produces films containing both PL and DG of various fractions, but a low temperature tends to favor a greater fraction of PL. The PL phase becomes the only discernible phase if thick films are shortly annealed at 230 °C.

Influence of Osmotic Pressure on Nanostructures in Thin Films of a Weakly-Segregated Block Copolymer and Its Blends with a Homopolymer

We studied the influence of osmotic pressure on nanostructures in thin films of a symmetric weakly-segregated polystyrene-block-poly (methyl methacrylate), P(S-b-MMA), block copolymer and its mixtures with a polystyrene (PS) homopolymer of various compositions. Thin films were deposited on substrates through surface neutralization. The surface neutralization results from the PS mats, which were oxidized and cross-linked by UV-light exposure. Thus, thermal annealing produced perpendicularly oriented lamellae and perforated layers, depending on the content of added PS chains. Nevertheless, a mixed orientation was obtained from cylinders in thin films, where a high content of PS was blended with the P(S-b-MMA). A combination of UV-light exposure and acetic acid rinsing was used to remove the PMMA block. Interestingly, the treatment of PMMA removal inevitably produced osmotic pressure and consequently resulted in surface wrinkling of perpendicular lamellae. As a result, a hierarchical structure with two periodicities was obtained for wrinkled films with perpendicular lamellae. The formation of surface wrinkling is due to the interplay between UV-light exposure and acetic acid rinsing. UV-light exposure resulted in different mechanical properties between the skin and the inner region of a film. Acetic acid rinsing produced osmotic pressure. It was found that surface wrinkling could be suppressed by reducing film thickness, increasing PS content and using high-molecular-weight P(S-b-MMA) BCPs.

Nanoscratch-Directed Self-Assembly of Block Copolymer Thin Films

Directed self-assembly (DSA) of block copolymer (BCP) thin films is of particular interest in nanoscience and nanotechnology due to its superior ability to form various well-aligned nanopatterns. Herein, nanoscratch-DSA is introduced as a simple and scalable DSA strategy allowing highly aligned BCP nanopatterns over a large area. A gentle scratching on the target substrate with a commercial diamond lapping film can form uniaxially aligned nanoscratches. As applied in BCP thin films, the nanoscratch effectively guides the self-assembly of overlying BCPs and provides highly aligned nanopatterns along the direction of the nanoscratch. The nanoscratch-DSA is not material-specific, allowing more versatile nanofabrication for various functional nanomaterials. In addition, we demonstrate that the nanoscratch-DSA can be utilized as a direction-controllable and area-selective nanofabrication method.

Effect of Molecular Architecture on the Self-Assembly of Bottlebrush Copolymers

The characteristics of a new architecture of bottlebrush copolymers (BBCPs) self-assembly were studied using self-consistent field theory. In this molecule, a series of AB linear diblock side chains were connected at the diblock junction using a C backbone. The control over the linker length and its chemical nature created an additional constraint on the intrinsic AB diblock microphase separation. Increasing side-chain crowding by increasing the grafting density and total degree of polymerization induced improved phase separation. This was reflected in the overall reduction in the effective interaction parameter between the diblocks as well as the abrupt increase in phase density when crossing the order-disorder transition. Side-chain crowding resulted in an increase in the equilibrium domain spacing compared to a linear diblock. On the other hand, the localization of block C at the AB interface generated a diffuse domain boundary and reduction in side-chain stretching. The unique behavior of BBCPs was observed in 1D confined systems where the molecule showed the natural tendency to orient domains parallel to neutral confinement in contrast to the behavior of confined diblocks. Such behavior largely depended on the degree of incompatibility between the AB blocks and BBCP structure. A ternary phase diagram was constructed for different proportions of each block. Rich morphologies of core-shell domains and tiling patterns were observed including octagonal and pentagonal polygons. The unique architecture of this bottlebrush molecule and its improved nanoscale properties make it an attractive candidate for various applications of nanotechnology.

Symmetric Diblock Copolymers in Cylindrical Confinement: A Way to Chiral Morphologies?

We investigate the confinement-induced formation and stability of helix morphologies in lamella-forming AB diblock copolymers via large-scale, particle-based, single-chain-in-mean-field simulations. Such helix structures are rarely observed in bulk or thin films. Structure formation is induced by quenching incompatibility, χN, from a disordered morphology. If the surfaces of the cylindrical confinement do not prefer one component over the other, we observe that stacked lamellae, with their normals along the cylinder axis, are the preferred morphology. Kinetically, this morphology initially forms close to the cylinder surface, whereas the spontaneous, spinodal microphase separation in the cylinder's interior gives rise to a microemulsion-like morphology, riddled with defects and no directional order. Subsequently, the ordered morphology on the cylinder surface progresses inward, pervading the entire volume. In case that the cylindrical pore is only partially filled, the additional confinement along the cylinder axis generally gives rise to incommensurability between the equilibrium spacing of stacked lamellae and the cylinder height. To accommodate this mismatch, the lamella normals will tilt away from the cylinder axis and generate helices of lamellae on the surface of the cylinder. Again, this order progresses from the cylinder surface inward, generating a chiral morphology. Because the spacing between the internal AB interfaces decreases upon approaching the helix center, the concomitant stress results in a decrease in the number of lamellae and the formation of unique dislocation defects. This type of chiral defect morphology is reproducibly formed by the kinetics of structure formation in partly filled cylindrical pores with nonpreferential surfaces and may find applications in photonic applications.

Temperature dependence of the conductivity of filled diblock copolymers

We demonstrate that an insulating diblock copolymer system (DBC) filled with conductive fillers can be used as an electrically responsive soft composite that changes its conductivity in response to temperature-induced changes in its morphology. By combining the phase field model describing the morphology of the DBC system with the Monte Carlo simulations and the resistor network model describing electrical properties of the filler network, we calculate the conductivity of this composite. The calculated conductivity is found to essentially depend, in particular, on the temperature of the composite. Changing the temperature is shown to result in morphological changes in the DBC system causing the structural changes in the filler network. In particular, the order-disorder transition in the host DBC system is found to be accompanied by the conductor-insulator transition in the filler network. The effect of the difference between the affinities of the fillers for dissimilar copolymer blocks on the composite conductivity, as well as the effect of the repulsive and attractive interaction between fillers, is considered in detail.

Oriented thick films of block copolymer made by multiple successive coatings: perforated lamellae versus oriented lamellae

Building 3D ordered nanostructures by copolymer deposition on a substrate implies a full control beyond the thin film regime. We have used here block copolymers (BCPs) forming bulk lamellar phases to form thick, i.e. much thicker than the lamellar period, structured films on a substrate. Films are formed by a simple method of multiple successive coatings. The film structure is controlled using the combined action of surface templating and annealing time. Sections of the thick layers were characterized by scanning electron microscopy (SEM) after etching of one of the BCP moieties. We show that perfect hexagonally perforated films (HPL) with lamellae parallel to the substrate are formed for a wide thickness range up to 300 nm. Grazing incidence small angle X-ray scattering (GISAXS) confirms such an organization by revealing that perforations sit on a hexagonal lattice. A lamellar organization perpendicular to the substrate is shown to take over for thicker films. A scenario consistent with our observations is proposed, where the sequence of phases results from the balance between surface and stretching energy effects.

Conductivity of Insulating Diblock Copolymer System Filled with Conductive Particles Having Different Affinities for Dissimilar Copolymer Blocks

We investigate the electrical response of the insulating diblock copolymer system (DBC) filled with conductive spherical fillers depending on the affinities of these fillers for copolymer blocks and the interaction between fillers. We demonstrate that the contrast (difference) between the affinities of the fillers for dissimilar copolymer blocks is a decisive factor that determines the distribution of these fillers in the DBC system. The distribution of filler particles, in turn, is found to be directly related to the electrical response of the DBC-particle composite. In particular, increasing the affinity contrast above a certain threshold value results in the insulator-conductor transition. This transition is found to be caused by the preferential localization of the fillers in the microphases of the DBC system having larger affinity for these fillers. The effect of the interaction between fillers is found to be secondary to the described effect of the affinity contrast that dominates in determining the distribution of fillers in the composite. This effect of the inter-particle interactions is shown to be significant only when the affinity contrast and filler volume fraction are sufficiently large.

Universal Interfacial Control through Polymeric Nanomosaic Coating for Block Copolymer Nanopatterning

The employment of self-assembly of soft materials has been accepted as an inexpensive, robust, and reliable patterning method. As their self-assembly relies on the delicate molecular interactions near the substrate, a precise prediction/control of the interface structure and dynamics is critical to achieve desired nanostructures. Herein, a polymeric nanomosaic (PNM) pattern is created from the air/water interfacial self-assembly of a block copolymer (BCP) and introduced as an effective interfacial energy control for substrates. As a demonstration, the PNM coating is employed to control the BCP film structures. The perpendicular orientation of BCP self-assembly, which requires neutral wetting conditions for both blocks, is difficult to achieve but can readily be obtained with the PNM coating upon a fine resolution of the pattern quality. The universal applicability of the PNM coating as an interfacial control has been confirmed on curved, flexible, and three-dimensional substrates. In addition, the PNM is introduced as an etching-free and reusable topcoat imparting free surface neutralization even for the high-χ BCP nanopatterning.

Single chain in mean field simulation of flexible and semiflexible polymers: comparison with discrete chain self-consistent field theory

Single chain in mean field (SCMF) simulation is a theoretical framework performing Monte Carlo moves of explicit polymer chains under quasi-instantaneously updated external fields which were originally imported from the self-consistent field theory (SCFT). Even though functional-based hybrid simulations are often used to compare the results of SCFT and MC simulation, the adoption of a finite number of coarse-grained segments makes direct comparison rather difficult. In this study, we perform SCMF simulation of block copolymers using various chain models and quantitatively compare it with discrete chain SCFT (DCSCFT) which finds the mean field solution of polymers with a finite number of segments. By comparing free energy and natural period of the symmetric block copolymer lamellar phase, we systematically show that DCSCFT serves as an intermediate step between SCMF simulation and SCFT. In addition, by adopting angle dependent bond potential, we perform SCMF simulation of semiflexible polymers using bead-spring and freely jointed chain models. As the chain stiffness increases, the lamellar phase tends to align perpendicular to the surfaces when confined between two neutral walls. We also investigate the effects of fluctuation and chain stiffness on the distribution of chain ends. The tendency of chain end segregation towards the surfaces turns out to increase as the chain stiffness increases for both homopolymer and block copolymer systems.

Field theoretic approach for block polymer melts: SCFT and FTS

This perspective addresses the development of polymer field theory for predicting the equilibrium phase behavior of block polymer melts. The approach is tailored to the high-molecular-weight limit, where universality reduces all systems to the standard Gaussian chain model, an incompressible melt of elastic threads interacting by contact forces. Using mathematical identities, this particle-based version of the model is converted to an equivalent field-based version that depends on fields rather than particle coordinates. The statistical mechanics of the field-based model is typically solved using the saddle-point approximation of self-consistent field theory (SCFT), which equates to mean field theory, but it can also be evaluated using field theoretic simulations (FTS). While SCFT has matured into one of the most successful theories in soft condensed matter, FTS are still in its infancy. The two main obstacles of FTS are the high computational cost and the occurrence of an ultraviolet divergence, but fortunately there has been recent groundbreaking progress on both fronts. As such, FTS are now well poised to become the method of choice for predicting fluctuation corrections to mean field theory.

100th Anniversary of Macromolecular Science Viewpoint: Block Copolymer Particles: Tuning Shape, Interfaces, and Morphology

Confined assembly of block copolymers (BCPs) is receiving increasing attention due to the ability to create unconventional morphologies that cannot be observed in the corresponding bulk systems. This effect is further driven by the simplicity and versatility of these procedures for controlling the shape of particles prepared by 3D soft confinement of BCPs in emulsions. By taking advantage of a mobile emulsion interface, the one-step formation of nonspherical BCP particles through spontaneous deformation is possible with design principles and theoretical models for controlling shape/nanostructure now being established. This Viewpoint highlights strategies for shape tuning of BCP particles, currently accessible shapes, their controllability, and potential application. The emergence of 3D soft confinement of BCPs and related theory is overviewed with a focus on current strategies, types of nonspherical shapes achieved, and structure-property relationships for nonspherical BCP particles. Finally, the applications and future perspectives for these materials are discussed.

Open-source code for self-consistent field theory calculations of block polymer phase behavior on graphics processing units

Self-consistent field theory (SCFT) is a powerful approach for computing the phase behavior of block polymers. We describe a fast version of the open-source Polymer Self-Consistent Field (PSCF) code that takes advantage of the massive parallelization provided by a graphical processing unit (GPU). Benchmarking double-precision calculations indicate up to 30× reduction in time to converge SCFT calculations of various diblock copolymer phases when compared to the Fortran CPU version of PSCF using the same algorithms, with the speed-up increasing with increasing unit cell size for the diblock polymer problems examined here. Where double-precision accuracy is not needed, single-precision calculations can provide speed-up of up to 60× in convergence time. These improvements in speed within an open-source format open up new vistas for SCFT-driven block polymer materials discovery by the community at large.

Strain-Dependent Nanowrinkle Confinement of Block Copolymers

This paper describes an all-soft, templated assembly of block copolymers (BCPs) with programmable alignment. Using polymeric nanowrinkles as a confining scaffold, poly(styrene)-block-poly(dimethylsiloxane) (PS-b-PDMS) BCPs were assembled to be parallel or perpendicular to the wrinkle orientation by manipulating the substrate strain. Self-consistent field theory modeling revealed that wrinkle curvature and surface affinity govern the BCP structural formation. Furthermore, control of BCP alignment was demonstrated for complex wrinkle geometries, various copolymer molecular weights, and functional wrinkle skin layers. This integration of BCP patterning with flexible 3D architectures offers a promising nanolithography approach for next-generation soft electronics.

High-Fidelity, Sub-5 nm Patterns from High-χ Block Copolymer Films with Vapor-Deposited Ultrathin, Cross-Linked Surface-Modification Layers

Despite their capability, sub-10 nm periodic nano-patterns formed by strongly segregating block copolymer (BCP) thin films cannot be easily oriented perpendicular to the substrate due to the huge surface energy differences of the constituent blocks. To produce perpendicular nano-patterns, the interfacial energies of both the substrate and free interfaces should be controlled precisely to induce non-preferential wetting. Unfortunately, high-performance surface modification layers are challenging to design, and different kinds of surface modification methods must be devised respectively for each neutral layer and top coat. Furthermore, conventional approaches, largely based on spin-coating processes, are highly prone to defect formation and may readily cause dewetting at sub-10 nm thickness. To date, these obstacles have hampered the development of high-fidelity, sub-5 nm BCP patterns. Herein, an all-vapor phase deposition approach initiated chemical vapor deposition is demonstrated to form 9-nm-thick, uniform neutral bottom layer and top coat with exquisite control of composition and thickness. These layers are employed in BCP films to produce perpendicular cylinders with a diameter of ≈4 nm that propagate throughout a BCP thickness of up to ≈60 nm, corresponding to five natural domain spacings of the BCP. Such a robust approach will serve as an advancement for the reliable generation of sub-10 nm nano-patterns.

Tunable helical structures formed by ABC triblock copolymers under cylindrical confinement

Block copolymers confined in nanopores provide unique achiral systems for the formation of helical structures. With AB diblock copolymers, stable single and double helical structures are observed. Aiming to obtain more different helical structures, we replace the AB diblock copolymer with linear ABC triblock copolymers. We speculate that a core-shell superstructure is formed within the nanopore, which is composed of a C-core cylinder wrapped by B-helices within the A-shell. Accordingly, the pore surface is set to be most attractive to the majority A-block and a typical set of interaction parameters is chosen as χACN ≪ χABN = χBCN = 80 to generate the frustrated interfaces. Furthermore, the volume fraction of B-block is fixed as fB = 0.1 to form helical cylinders. A number of helical structures with strands ranging from 1 to 5 are predicted by self-consistent field theory, and in general, the number of strands decreases as the volume fraction of C-block fC increases in a given nanopore. More surprisingly, the variation of helical strand in the confined system has an opposite trend to that in the bulk, which mainly results from the constraint of the cylindrical confinement on the change of the curvature between the outer A-layer and the inner B/C-superdomain. Our work demonstrates a facile way to fabricate different helical superstructures.

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.

Mesoscopic theory for systems with competing interactions near a confining wall

Mesoscopic theory for self-assembling systems near a planar confining surface is developed. Euler-Lagrange equations and the boundary conditions (BCs) for the local volume fraction and the correlation function are derived from the density functional theory expression for the grand thermodynamic potential. Various levels of approximation can be considered for the obtained equations. The lowest-order nontrivial approximation [generic model (GM)] resembles the Landau-Brazovskii-type theory for a semi-infinite system. Unlike in the original phenomenological theory, however, all coefficients in our equations and BCs are expressed in terms of the interaction potential and the thermodynamic state. Analytical solutions of the linearized equations in the GM are presented and discussed on a general level and for a particular example of the double-Yukawa potential. We show exponentially damped oscillations of the volume fraction and the correlation function in the direction perpendicular to the confining surface. The correlations show oscillatory decay in directions parallel to this surface too, with the decay length increasing significantly when the system boundary is approached. The framework of our theory allows for a systematic improvement of the accuracy of the results.

The influence of curvature on domain distribution in binary mixture membranes

Curvature-induced domain sorting, a strategy exploited by cells to organize membrane components, is a promising mechanism to control self-assembly of materials. To understand this phenomenon, this work explores the effects of curvature on component rearrangement in thin polymer films and lipid bilayers supported on sinusoidal substrates. Specifically, self-consistent field theory (SCFT) was used to study the spatial distribution of polymers in blends containing conformationally asymmetric chains. In addition, coarse-grained molecular dynamics (MD) simulations were used to probe the arrangement of rigid lipid domains in a relatively soft lipid matrix. Besides the expected preference of rigid species localizing in regions with low mean curvature, both systems exhibit unexpected localization of rigid components in comparatively high curvature regions. The origins of this unexpected sorting are discussed in terms of entropic and enthalpic contributions. In summary, this study demonstrates that domain distribution strongly depends on local topography and further highlights the collective effects that thermodynamic forces have on the morphological behavior of membranes.

Numerical implementation of pseudo-spectral method in self-consistent mean field theory for discrete polymer chains

In the standard self-consistent field theory (SCFT), a polymer chain is modeled as an infinitely flexible Gaussian chain, and the partition function is calculated by solving a differential equation in the form of a modified diffusion equation. The Gaussian chain assumption makes the standard SCFT inappropriate for modeling of short polymers, and the discrete chain SCFT in which the partition function is obtained through recursive integrals has recently been suggested as an alternative method. However, the shape of the partition function integral makes this method much slower than the standard SCFT when calculated in the real space. In this paper, we implement the pseudospectral method for the discrete chain SCFT adopting the bead-spring or freely jointed chain (FJC) model, and a few issues such as the accurate discretization of the FJC bond function are settled in this process. With the adoption of the pseudospectral method, our calculation becomes as fast as that of the standard SCFT. The integral equation introduces a new boundary condition, the neutral boundary, which is not available in the standard SCFT solving the differential equation. This interesting physical situation is combined with the finite-range interaction model for the study of symmetric block copolymers within thin films. We find that the surface-perpendicular block copolymer lamellar phase becomes preferable to the surface-parallel one when both the top and bottom surfaces are neutral.

Reactive block copolymers for patterned surface immobilization with sub-30 nm spacing

Reactive polystyrene-block-polyisoprene copolymers are synthesized by nitroxide-mediated polymerization, self-assemble within ultra-thin films, and exhibit surface reactivity for patterned immobilization.

Thermal Approaches to Perpendicular Block Copolymer Microdomains in Thin Films: A Review and Appraisal

Block copolymer thin films are highly versatile and accessible materials capable of producing nanofeatures in the size regime of a few to hundreds of nanometers by a simple spin-coating-and-anneal process. Unfortunately, this simple protocol usually leads to parallel microdomains, which limits the applicability of such nanofeatures. A great deal of effort has been put into achieving perpendicular microdomains, but those that incorporate thermal annealing are arguably the most practical and reproducible in the lab and industry. This review discusses the recent ongoing efforts on various thermal approaches to achieving perpendicular microdomains in order to provide the readers with a toolbox to work with.

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.