Symmetric diblock copolymers in nanopores: Monte Carlo simulations and strong-stretching theory

Controlling the Formation of Polyhedral Block Copolymer Nanoparticles: Insights from Process Variables and Dynamic Modeling

This study delves into the formation of nanoscale polyhedral block copolymer particles (PBCPs) exhibiting cubic, octahedral, and variant geometries. These structures represent a pioneering class that has never been fabricated previously. PBCP features distinct variations in curvature on the outer surface, aligning with the edges and corners of polyhedral shapes. This characteristic sharply contrasts with previous block copolymers (BCPs), which displayed a smooth spherical surface. The emergence of these cornered morphologies presents an intriguing and counterintuitive phenomenon and is linked to process parameters, such as evaporation rates and initial concentration, while keeping other variables constant. Using a system of coupled Cahn-Hillard (CCH) equations, we uncover the mechanisms driving polyhedral particle formation, emphasizing the importance of controlling relaxation parameters for shape variable u and microphase separation v. This unconventional approach, differing from traditional steepest descent method, allows for precise control and diverse polyhedral particle generation. Accelerating the shape variable u proves crucial for expediting precipitation and aligns with experimental observations. Employing the above theoretical model, we achieve shape predictions for particles and the microphase separation within them, which overcomes the limitations of ab initio computations. Additionally, a numerical stability analysis discerns the transient nature versus local minimizer characteristics. Overall, our findings contribute to understanding the complex interplay between process variables and the morphology of polyhedral BCP nanoparticles.

Semicrystalline Block Copolymers in Rigid Confining Nanopores

We have investigated PLLA crystallization in lamellae-forming PS-b-PLLA confined to straight cylindrical nanopores under weak confinement (nanopore diameter D/equilibrium PS-b-PLLA period L0 ≥ 4.8). Molten PS-b-PLLA predominantly forms concentric lamellae along the nanopores, but intertwined helices occur even for D/L0 ≈ 7.3. Quenching PS-b-PLLA melts below TG(PS) results in PLLA cold crystallization strictly confined by the vitrified PS domains. Above TG(PS), PLLA crystallization is templated by the PS-b-PLLA melt domain structure in the nanopore centers, while adsorption on the nanopore walls stabilizes the outermost cylindrical PS-b-PLLA shell. In between, the nanoscopic PS-b-PLLA melt domain structure apparently ripens to reduce frustrations transmitted from the outermost immobilized PS-b-PLLA layer. The onset of PLLA crystallization catalyzes the ripening while transient ripening states are arrested by advancing PLLA crystallization. Certain helical structure motifs persist PLLA crystallization even if PS is soft. The direction of fastest PLLA crystal growth is preferentially aligned with the nanopore axes to the same degree as for PLLA homopolymer, independent of whether PS is vitreous or soft.

Order-order transitions of diblock copolymer melts under cylindrical confinement

The self-assembly behavior of AB diblock copolymers under cylindrical confinement is investigated using the self-consistent field theory. We focus on the impact of the confinement on the order-order transitions of three-dimensional morphologies by constructing two types of phase diagrams with continuously varying block compositions. One type is with respect to the block composition and the immiscibility parameter for various pore sizes, in which the order-order transitions are shown to be strongly impacted by the pore curvature and thus largely different from the bulk ones. Note that the morphologies are categorized by the intrinsical geometry of their domains, i.e., that helical morphologies are regarded as one type of cylindrical phase. Another type of phase diagram is with respect to the block composition and the pore diameter, which exhibits a number of interesting order-order transitions, especially the transition sequence from a straight line of spheres, to one straight cylinder and stacked disks as the pore diameter increases. A critical point is observed at which the stability region of the straight cylinder vanishes and thereby the spheres transform into the stacked disks continuously. The mechanism of these phase transitions is rationalized in the context of the bulk factors as well as an additional factor, i.e., the competition between the spontaneous curvature of the copolymer and the imposed curvature by the nanopore.

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.

Self-assembly of diblock copolymer confined in an array-structure space

The combination of top-down and bottom-up technologies is an effective method to create the novel nanostructures with long range order in the field of advanced materials manufacture. In this work, we employed a polymeric self-consistent field theory to investigate the pattern formation of diblock copolymer in a 2D confinement system designed by filling pillar arrays with various 2D shapes such as squares, rectangles, and triangles. Our simulation shows that in such confinement system, the microphase structure of diblock copolymer strongly depends on the pitch, shape, size, and rotation of the pillar as well as the surface field of confinement. The array structures can not only induce the formation of new phase patterns but also control the location and orientation of pattern structures. Finally, several methods to tune the commensuration and frustration of array-structure confinement are proposed and examined.

Patchy nanoparticles self-assembled from linear triblock copolymers under spherical confinement: a simulated annealing study

The self-assembly of linear ABC triblock copolymers confined in spherical nanopores is studied using a simulated annealing technique. Morphological phase diagrams as a function of the pore diameter, the selectivity of the pore-wall to the terminal blocks, and the copolymer composition are constructed. A variety of patchy nanoparticles and multiple morphological transitions are identified. Janus nanoparticles, which can be regarded as particles with one patch, are observed inside small nanopores. With increasing the pore diameter, the number of patches on a nanoparticle surface increases from one to two, four, five, six, and seven. The size of each patch increases periodically. The number of patches also increases with increasing the wall selectivity. The distribution of the patches on the surface of a given particle is highly symmetric. The interior structures of the patchy nanoparticles and the morphological transition are investigated by calculating the bridging fraction, the mean square end-to-end distance and the average contact number between different components. A series of entropy-driven morphological transitions is predicted. Furthermore, it is found that the overall patchy morphology is largely controlled by the volume fraction of the middle B-block, while the internal structure is largely controlled by the volume fraction ratio of the two terminal blocks. Our study demonstrates that the size of nanopores, the pore-wall selectivity, and the copolymer composition could be utilized as effective means to tune the structure and properties of the anisotropic nanoparticles.

Segmented helical structures formed by ABC star copolymers in nanopores

Self-assembly of ABC star triblock copolymers confined in cylindrical nanopores is studied using self-consistent mean-field theory. With an ABC terpolymer forming hexagonally-arranged cylinders, segmented into alternative B and C domains, in the bulk, we observe the formation in the nanopore of a segmented single circular and non-circular cylinder, a segmented single-helix, and a segmented double-helix as stable phases, and a metastable stacked-disk phase with fourfold symmetry. The phase sequence from single-cylinder, to single-helix, and then to double-helix, is similar as that in the cylindrically-confined diblock copolymers except for the absence of an equilibrium stacked-disk phase. It is revealed that the arrangement of the three-arm junctions plays a critical role for the structure formation. One of the most interesting features in the helical structures is that there are two periods: the period of the B/C domains in the helix and the helical period. We demonstrate that the period numbers of the B/C domains contained in each helical period can be tuned by varying the pore diameter. In addition, it is predicted that the period number of B/C domains can be any rational in real helical structures whose helical period can be tuned freely.

Soft confinement-induced morphologies of diblock copolymers

The self-assembly of diblock copolymers under soft confinement is studied systematically using a simulated annealing method applied to a lattice model of polymers. The soft confinement is realized by the formation of polymer droplets in a poor solvent environment. Multiple sequences of soft confinement-induced copolymer aggregates with different shapes and self-assembled internal morphologies are predicted as functions of solvent-polymer interaction and the monomer concentration. It is discovered that the self-assembled internal morphology of the aggregates is largely controlled by a competition between the bulk morphology of the copolymer and the solvent-polymer interaction, and the shape of the aggregates can be non-spherical when the internal morphology is anisotropic and the solvent-polymer interaction is weak. These results demonstrate that droplets of diblock copolymers formed in poor solvents can be used as a model system to study the self-assembly of copolymers under soft confinement.

Geometric frustration phases of diblock copolymers in nanoparticles

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

Continuous concentric lamellar block copolymer nanofibers with long range order

Fibers with long-range ordered internal structures have applications in various areas such as photonic band gap fibers, optical waveguides, wearable power, sensors, and sustained drug release. Up to now, such fibers have been formed by melt extrusion or drawing from a macroscopic preformed rod and were typically limited to diameters >10 microm with internal features >1 microm (Abouraddy, A. F.; et al. Nat. Mater. 2007, 6, 336). We describe a new class of continuous fibers and fibrous membranes with long-range ordered concentric lamellar structure that have fiber diameters and feature sizes 2-3 orders of magnitude smaller than those made by conventional methods. These fibers are created through confined self-assembly of block copolymers within core-shell electrospun filaments. In contrast to the copolymer in bulk or thin films, the domains of the concentric lamellar structure are shown here to vary quantitatively with (radial) position and to exhibit a novel dislocation that accommodates variations in fiber diameter robustly, permitting for the first time the realization of long-range order in technologically meaningful, continuous fibers with approximately 300 nm diameter and 50 nm radial period.

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.