From Cylinders to Helices upon Confinement

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.

Confinement-induced self-assembly of a diblock copolymer within a non-uniform cylindrical nanopore

The self-assembly of a diblock copolymer melt confined within a non-uniform cylindrical nanopore is studied using the self-consistent field theory. The non-uniformity manifests in the form of a converging-diverging cylindrical nanopore. The axial variation in pore diameter presents a range of curvatures within the same confinement pore as opposed to a single curvature in a uniform-diameter cylindrical pore. The introduction of multiple curvatures leads to the formation of novel microstructures not accessible in uniform cylindrical confinement. The well-known equilibrium structures like a single helix, double helices, and concentric lamella under cylindrical confinement transition into new morphologies such as hyperboloidal phases, microstructures containing rings with a bead, rings with spheres, and a squeezed helical phase as the pore diameter varies axially. The converging-diverging geometry of the confining pore renders the helical phases seen in the cylindrical pore less favorable. A phase diagram in the parametric space of the block fraction and the ratio of the smallest and largest pore radii has been constructed to depict the order-order transition of various microstructures. The ratio of radii, a measure of the non-uniformity of the pore, along with the pore length brings out some interesting morphologies. The mechanism of these structural transitions is understood as the interplay between the variation in pore curvature attributed to the non-uniformity, the spontaneous curvature of the block copolymer interface, and the enthalpic interaction between the segregated blocks.

Orbital Angular Momentum from Self-Assembled Concentric Nanoparticle Rings

Ring-shaped nanostructures can focus, filter, and manipulate electromagnetic waves, but are challenging to incorporate into devices using standard nanofabrication techniques. Directed self-assembly (DSA) of block copolymers (BCPs) on lithographically patterned templates has successfully been used to fabricate concentric rings and spirals as etching masks. However, this method is limited by BCP phase behavior and material selection. Here, a straightforward approach to generate ring-shaped nanoparticle assemblies in thin films of supramolecular nanocomposites is demonstrated. DSA is used to guide the formation of concentric rings with radii spanning 150-1150 nm and ring widths spanning 30-60 nm. When plasmonic nanoparticles are used, ring nanodevice arrays can be fabricated in one step, and the completed devices produce high-quality orbital angular momentum (OAM). Nanocomposite DSA simplifies and streamlines nanofabrication by producing metal structures without etching or deposition steps; it also introduces interparticle coupling as a new design axis. Detailed analysis of the nanoparticle ring assemblies confirms that the supramolecular system self-regulates the spatial distribution of its components, and thus exhibits a degree of flexibility absent in DSA of BCPs alone, where structures are determined by polymer-pattern incommensurability. The present studies also provide guidelines for developing self-regulating DSA as an alternative to incommensurability-driven methods.

Self-organization of a 4-miktoarm star block copolymer induced by cylindrical confinement

Self-consistent field calculations have been carried out to reveal the self-assembly behavior of a melt of the ABCD star tetrablock copolymer confined within a cylindrical nanopore. The miktoarm star block copolymer exhibits a rich self-assembly behavior with a myriad of interesting three-dimensional ordered phases with the potential to produce advanced nanomaterials. The broad array of ordered mesophases includes helical microstructures, stack of rings/doughnuts, honeycomb structure, and perforated lamella with beads, depending on the individual block fractions and the size of the cylindrical nanopore. Such chiral motifs generated from achiral polymeric molecules are fascinating due to their superior performance in sophisticated opto-electronic devices. The study also demonstrates an interesting morphology, viz. a honeycomb structure, obtained from the self-organization of ABCD star block copolymer molecules with equal block fractions. The system exhibits order-order phase transition covering a range of ordered morphologies by changing either the block fraction or the nanopore radius. A representative phase diagram in terms of block fractions is constructed. These novel ordered microstructures, arising mainly out of structural frustration and confinement-induced entropy loss, can serve as structural scaffolds to host the spatial distribution of nanoparticles resulting into novel nanocomposites with significantly enhanced as well as controllable properties.

Controlling the chirality and number of strands of helices self-assembled from achiral block copolymers confined inside a nanopore: a simulation study

Achiral block copolymers can self-assemble into helical structures when confined inside a cylindrical nanopore. However, controlling the chirality and the number of strands of helices is challenging. We present our simulation results of the influence of a chiral patch added to the confining nanopore on the structures and chirality of helices self-assembled from achiral cylinder-forming diblock copolymers under the confinement. Our results indicate that, when the designed patch is of proper geometry, it can induce the formation of helical structures and exhibit good control over their chirality. The bottom surface of the patch can induce the formation of a characteristic local structure near and parallel to it. It is the characteristic local structure that directs the formation of helices and of their chirality consistent with that of the patch. A large patch angle or the top/bottom surface of a weakly selective pore promotes the formation of double-helices compared to single-helices by enlarging the pitch of the helices near the patch or through the entropic attraction of the top surface of the pore to the minority blocks.

Self-Assembly of Polymer Grafted Nanoparticles within Spherically Confined Diblock Copolymers

Geometric confinement plays an important role in the generation of interesting microstructures on account of structural frustration and confinement-induced entropy loss. In the present study, self-consistent field calculations have been performed to examine the self-assembly behavior of a mixture of diblock copolymers and polymer grafted nanoparticles within a spherical confinement. The analysis is aimed at obtaining the equilibrium distribution of nanoparticles with a high degree of order. The ordered mesophases of diblock copolymers provide useful templates to achieve ordering of nanoparticles in a selective domain. Self-assembly of nanoparticles within frustrated diblock copolymers is found to be very different from the bulk. A rich variety of equilibrium morphologies are observed depending on the degree of confinement and the extent of particle loading. In addition, the role of particle size and selectivity along with the length and the number of polymer chains grafted onto the surface of nanoparticles are analyzed to understand the self-assembly behavior. The specific interest is to obtain the chiral structures out of achiral block copolymers subjected to spherical confinement. The realization of various captivating microstructures, such as chiral ordering of nanoparticles, is highly essential to produce advanced nanomaterials with superior physical properties.

Self-Directed Self-Assembly of 3D Tailored Block Copolymer Nanostructures

Directed self-assembly (DSA) of block copolymers (BCPs) provides a powerful tool to fabricate various 2D nanostructures. However, it still remains a challenge to extend DSA to make uniform and complex 3D nanostructures through BCP self-assembly. In this paper, we introduce a method to fabricate various nanostructures in 3D and test it using simulations. In particular, we employ dissipative particle dynamics (DPD) simulation to demonstrate that uniform multilayer nanostructures can be achieved by alternating the stacking of two "orthogonal" BCPs films, AB copolymer film and AC copolymer film, without the need to cross-link or etch any of the components. The assembly of a new layer occurs on top of the previous bottom layer, and thus the structural information from the substrate is propagated upward in the film, a process we refer to as self-directed self-assembly (SDSA). If this process is repeated many times, one can have tailored multilayer nanostructures. Furthermore, the natural (bulk) phases of the block copolymers in each layer do not need to be the same, so one can achieve complex 3D assemblies that are not possible with a single-phase 3D system. This method in conjunction with grapho (or chemo) epitaxy is able to evolve a surface pattern into a 3D nanostructure. Here we show several examples of nanostructures fabricated by this process, which include aligned cylinders, spheres on top of cylinders, and orthogonal nanomeshes. Our work should be useful for creating complex 3D nanostructures using self-assembly.

Diblock copolymer templated self-assembly of grafted nanoparticles under circular pore confinement

Geometrical confinement plays an important role in generating novel molecular organization arising out of structural frustration and confinement-induced entropy loss. In the present study, we perform self-consistent mean-field theoretical calculations to examine a mixture of a diblock copolymer and polymer grafted nanoparticles confined in a cylindrical nanopore. The two-dimensional analysis is aimed at constructing the equilibrium nanostructures decorated with particles in an ordered manner. The rich variety of ordered mesophases of the diblock copolymer under confinement provide a template to achieve the self-assembly of nanoparticles in a selective domain. The localization behavior of nanoparticles under confinement is found to be qualitatively different from that in a bulk system. In particular, for the concentric lamellar phase the particles tend to localize predominantly in the region of greater curvature within the curved lamella. The incorporation of grafted nanoparticles also results in a transition in ordered phases. Various equilibrium morphologies are observed depending upon the degree of confinement, particle loading, density of grafted segments and selectivity of the particle core to the polymeric species. The ordering of particles and the ensuing equilibrium nanostructures are analyzed. The comprehensive understanding of the self-assembly behavior of particles enables one to design novel nanomaterials with desirable material properties.

Hierarchical self-assembly of a PS-b-P4VP/PS-b-PNIPAM mixture into multicompartment micelles and their response to two-dimensional confinement

Finely tuned synergistic effects among different blocks could realize intriguing hierarchical self-assembly of block copolymers and such hierarchical self-assembly could be manipulated by cylindrical confinement to tune the structures of assemblies.

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.

Diverse chiral assemblies of nanoparticles directed by achiral block copolymers via nanochannel confinement

It is a challenging task to realize large-area manufacture of chiral geometries of nanoparticles in solid-state materials, which exhibit strongly chiroptical responses in the visible and near-infrared ranges. Herein, novel nanocomposites, made from mixtures of achiral block copolymers and nanoparticles in a geometrically confined environment, are conceptually proposed to construct the chiral assemblies of nanoparticles through a joint theoretical-calculation framework and experimental discussion. It is found that the nanochannel-confined block copolymers self-assemble into a family of intrinsically chiral architectures, which serve as structural scaffolds to direct the chiral arrangement of nanoparticles. Through calculations of chiral order parameters and simulations of discrete dipole approximation, it is further demonstrated that certain members of this family of nanoparticle assemblies exhibit intense chiroptical activity, which can be tailored by the nanochannel radius and the nanoparticle loading. These findings highlight the multiple levels of structural control over a class of chiral assemblies of nanoparticles and the functionalities of emerging materials via careful design and selection of nanocomposites.

Formation of homochiral helical nanostructures in diblock copolymers under the confinement of nanopores

The control of the homochirality of helical structures formed in achiral systems is of great interest as it is helpful for understanding the origin of homochirality in life.

Microwave-annealing-induced nanowetting of block copolymers in cylindrical nanopores

Block copolymers have attracted great attention because of their abilities to self-assemble into well-ordered microphase-separated structures. To generate nanopatterns of block copolymers with long-range ordering and low-defect densities in shorter time scales, microwave annealing has recently been applied. Microwave annealing, however, has so far only been used for block copolymer bulks and thin films. In this work, we discover that microwave annealing can be successfully applied to three-dimensional block copolymer nanostructures by studying the infiltration and microphase separation of block copolymers in cylindrical nanopores upon microwave irradiation. Cylinder-forming and lamella-forming poly(styrene-block-dimethylsiloxane) (PS-b-PDMS) are introduced into the nanopores of anodic aluminum oxide (AAO) templates. In addition, AAO templates with different pore sizes are used to study the effect of the commensurabilities between the pore diameters and the repeating periods of the block copolymers on the morphologies of the block copolymer nanostructures.

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.

Blending Homopolymers for Controlling the Morphology Transitions of Block Copolymer Nanorods Confined in Cylindrical Nanopores

The microphase separation of block copolymers in confined geometries has been widely investigated over the last few decades. The controllability and versatility of the confinement-induced morphologies, however, are still difficult to be achieved because of the limited experimental parameters in the process of fabricating the confined nanostructures. In this work, we study the morphology transitions of lamellae-forming polystyrene-block-polydimethylsiloxane (PS-b-PDMS) nanorods confined in the nanopores of anodic aluminum oxide (AAO) templates. The nanorods are formed by solvent-assisted template wetting, and the morphologies are compared to those in the bulk state. By blending PS-b-PDMS with homopolystyrene (hPS), the morphologies of the nanorods can be controlled because of the changes of the effective volume fractions. Special morphology transitions from concentric lamellar morphology, to multihelical morphology, and finally to spherical-like morphology are observed by increasing the weight ratios of hPS. hPS with different molecular weights is also applied to investigate the effect of hPS on the morphologies of the PS-b-PDMS/hPS blend nanostructures. The unusual morphologies are further confirmed by a selective removal process, which also generates nanochannels for possible refilling with functional materials.

Selective solvent-induced reconstruction in confined space: one-dimensional mesoporous block copolymer structures in cylindrical nanopores

The formation of polymer nanostructures confined in cylindrical nanopores via a novel selective solvent-induced reconstruction process is investigated.

Rich Variety of Three-Dimensional Nanostructures Enabled by Geometrically Constraining Star-like Block Copolymers

The influence of star-like architecture on phase behavior of star-like block copolymer under cylindrical confinement differs largely from the bulk (i.e., nonconfinement). A set of intriguing self-assembled morphologies and the corresponding phase diagrams of star-like (AB)f diblock copolymers with different numbers of arms f (i.e., f = 3, 9, 15, and 21) in four scenarios (ϕA = 0.3 and V0 > 0; ϕA = 0.3 and V0 < 0; ϕA = 0.7 and V0 > 0; and ϕA = 0.7 and V0 < 0 (where ϕA is the volume fraction of A block) and V0 < 0 and V0 > 0 represent that the pore wall of cylindrical confinement prefers the inner A block (i.e., A-preferential) and B block (i.e., B-preferential), respectively) were for the first time scrutinized by employing the pseudospectral method of self-consistent mean-field theory. Surprisingly, a new nanoscopic phase, that is, perforated-lamellae-on-cylinder (denoted PC), was observed in star-like (AB)3 diblock copolymer at ϕA = 0.3 and V0 > 0. With a further increase in f, a single lamellae (denoted L1) was found to possess a larger phase region. Under the confinement of A-preferential wall (i.e., V0 < 0) at ϕA = 0.3, PC phase became metastable and its free energy increased as f increased. Quite intriguingly, when ϕA = 0.7 and V0 > 0, where an inverted cylinder was formed in bulk, the PC phase became stable, and its free energy decreased as f increased, suggesting the propensity to form PC phase under this condition. Moreover, in stark contrast to the phase transition of C1 → L1 → PC (C1, a single cylindrical microdmain) at ϕA = 0.3 and V0 > 0, when subjected to the A-preferential wall (ϕA = 0.7), a different phase transition sequence (i.e., C1 → PC → L1) was identified due to the formation of a double-layer structure. On the basis of our calculations, the influence of star-like architecture on (AB)f diblock copolymer under the imposed cylindrical confinement, particularly the shift of the phase boundaries as a function of f, was thoroughly understood. These self-assembled nanostructures may hold the promise for applications as lithographic templates for nanowires, photonic crystals, and nanotechnology.

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.

Chiral Nanoarchitectonics: Towards the Design, Self-Assembly, and Function of Nanoscale Chiral Twists and Helices

Helical structures such as double helical DNA and the α-helical proteins found in biological systems are among the most beautiful natural structures. Chiral nanoarchitectonics, which is used here to describe the hierarchical formation and fabrication of chiral nanoarchitectures that can be observed by atomic force microscopy (AFM), scanning tunneling microscopy (STM), scanning electron microscopy (SEM), or transmission electron microscopy (TEM), is one of the most effective ways to mimic those natural chiral nanostructures. This article focuses on the formation, structure, and function of the most common chiral nanoarchitectures: nanoscale chiral twists and helices. The types of molecules that can be designed and how they can form hierarchical chiral nanoarchitectures are explored. In addition, new and unique functions such as amplified chiral sensing, chiral separation, biological effects, and circularly polarized luminescence associated with the chiral nanoarchitectures are discussed.

Non-destructive means of probing a composite polyamide membrane for characteristic free volume, void, and chemical composition

Positron annihilation spectroscopy measures free volume in membranes at the sub-nanometer scale (0.1–1 nm).

Probing the Effect of Molecular Nonuniformity in Directed Self-Assembly of Diblock Copolymers in Nanoconfined Space

Various complex self-assembled morphologies of lamellar- and cylinder-forming block copolymers comprising poly(dimethylsiloxane)-b-polylactide (PDMS-b-PLA) confined in cylindrical channels were generated. Combining top-down lithography with bottom-up block copolymer self-assembly grants access to morphologies that are otherwise inaccessible with the bulk materials. Channel diameter (D) was systematically varied with four diblock copolymers having different compositions and bulk domain spacing (L0), corresponding to a range of frustration ratios (D/L0 from 2 to 4). Excessive packing frustration imposed by the channels leads to contorted domains. The resulting morphologies depend strongly on both D/L0 and copolymer composition. Under several circumstances, mixtures of complex morphologies were observed, which hypothetically arise from the severe sensitivity to D/L0 combined with the inherent compositional/molar mass dispersities associated with the nonuniform synthetic materials and silicon templates. Stochastic calculations offer compelling support for the hypothesis, and tractable pathways toward solving this apparent conundrum are proposed. The materials hold great promise for next-generation nanofabrication to address several emerging technologies, offering significantly enhanced versatility to basic diblock copolymers as templates for fabricating complex nanoscale objects.

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.

Templated self-assembly of block copolymers and morphology transformation driven by the Rayleigh instability

In the current study, we investigate the self-assembly of polystyrene-block-poly(4-vinylpyridine) (PS-b-P4VP) confined in the nanopores of the anodic aluminum oxide (AAO) template and the subsequent morphology transformation induced by the Rayleigh instability. PS-b-P4VP nanotubes and nanorods with various internal nanostructures are fabricated by wetting the AAO template with PS-b-P4VP/chloroform solution, and then followed by solvent evaporation. After the removal of AAO template by potassium hydroxide solution, several different solvents (chloroform, toluene, and N,N-dimethylformamide) with different qualities are used to swell and anneal those nanotubes and nanorods suspended in aqueous media. Morphology transformation from nanostructured PS-b-P4VP nanotubes or nanorods to ordered nanospheres is observed by annealing upon chloroform and toluene while the morphology remains unchanged upon N,N-dimethylformamide annealing, indicating that solvent quality is a key factor in tuning the morphology and internal structures. Kinetics study and theoretical analysis for the morphology transition from two-dimensional (2D) block copolymer (BCP) nanotubes and nanorods to three-dimensional (3D) BCP nanospheres are further performed. From the morphological evolution and the quantitative calculation, it is confirmed that this transition is induced by the Rayleigh instability. This study provides a simple but promising method, that is, solvent annealing method, for the fabrication of BCP nanospheres with ordered internal nanostructures, which may have great application in drug delivery and other nanotechnology.

Honeycombs in honeycombs: complex liquid crystal alumina composite mesostructures

Small-angle X-ray scattering (SAXS) and atomic force microscopy (AFM) were used to study orientation patterns of two polyphilic liquid crystals (LC) confined to cylindrical pores of anodic aluminum oxide (AAO). The hierarchical hybrid systems had the LC honeycomb (lattice parameter 3.5-4 nm) inside the pores of the AAO honeycomb (diameters 60 and 400 nm). By conducting complete reciprocal space mapping using SAXS, we conclude that the columns of both compounds align in planes normal to the AAO pore axis, with a specific crystallographic direction of the LC lattice aligning strictly parallel to the pore axis. AFM of LC-containing AAO fracture surfaces further revealed that the columns of the planar anchoring LC (compound 1) formed concentric circles in the plane normal to the pore axis near the AAO wall. Toward the pore center, the circles become anisometric "racetrack" loops consisting of two straight segments and two semicircles. This mode compensates for slight ellipticity of the pore cross section. Indications are, however, that for perfectly circular pores, circular shape is maintained right to the center of the pore, the radius coming down to the size of a molecule. For the homeotropically anchoring compound 2, the columns are to the most part straight and parallel to each other, arranged in layers normal to the AAO pore axis, like logs in an ordered pile. Only near the pore wall the columns splay somewhat. In both cases, columns are confined to layers strictly perpendicular to the AAO pore axis, and there is no sign of escape to the third dimension or of axial orientation, the latter having been reported previously for some discotic LCs. The main cause of the two new LC configurations, the "racetrack" and the "logpile", and of their difference from those of confined nematic LC, is the very high splay energy and low bend energy of columnar phases.

Self-consistent field theory of block copolymers on a general curved surface

In this work, we propose a theoretical framework based on the self-consistent field theory (SCFT) for the study of self-assembling block copolymers on a general curved surface. Relevant numerical algorithms are also developed. To demonstrate the power of the approach, we calculate the self-assembled patterns of diblock copolymers on three distinct curved surfaces with different genus. We specially study the geometrical effects of curved surfaces on the conformation of polymer chains as well as on the pattern formation of block copolymers. By carefully examining the diffusion equation of the propagator on curved surfaces, it is predicted that Gaussian chains are completely unaware of the extrinsic curvature but that they will respond to the intrinsic curvature of the surface. This theoretical assertion is consistent with our SCFT simulations of block copolymers on general curved surfaces.