Atomically Flat, 2D Edge-Directed Self-Assembly of Block Copolymers.

Nanoscale shape engineering is an essential requirement for the practical use of 2D materials, aiming at precisely customizing optimal structures and properties. In this work, sub-10-nm-scale block copolymer (BCP) self-assembled nanopatterns finely aligned along the atomic edge of 2D flakes, including graphene, MoS2 , and h-BN, are exploited for reliable nanopatterning of 2D materials. The underlying mechanism for the alignment of the self-assembled nanodomains is elucidated based on the wetting layer alternation of the BCP film in the presence of intermediate 2D flakes. The resultant highly aligned nanocylinder templates with remarkably low levels of line edge roughness (LER) and line-width roughness (LWR) yield a sub-10-nm-wide graphene nanoribbon (GNR) array with noticeable switching characteristics (on-to-off ratio up to ≈6 × 104 ).

The Boltzmann fair division for distributive justice.

Fair division is a significant, long-standing problem and is closely related to social and economic justice. The conventional division methods such as cut-and-choose are hardly applicable to real-world problems because of their complexity and unrealistic assumptions about human behaviors. Here we propose a fair division method from a completely different perspective, using the Boltzmann division. The mathematical model of the Boltzmann division was developed for both homogeneous and heterogeneous cake-cutting problems, and the realistic human factors (contributions, needs, and preferences) of the multiple participating players could be successfully integrated. The Boltzmann division was then optimized by maximizing the players' total utility. We show that the Boltzmann fair division is a division method favorable to the socially disadvantaged or underprivileged, and it is drastically simple yet highly versatile and can be easily fine-tuned to directly apply to a variety of social, economic, and political division problems.

Graphoepitaxy of Symmetric Six-Arm Star-Shaped Poly(methyl methacrylate)-block-Polystyrene Copolymer Thin Film.

The authors perform directed self-assembly based on graphoepitaxy of symmetric six-arm star-shaped poly(methyl methacrylate)-block-polystyrene copolymer [(PMMA-b-PS)6 ] thin film. The affinity between each block and the trench wall is adjusted by using polymer brushes or selective gold (Au) deposition. When the surface of the trench is strongly selective for the PMMA block, (n+0.75)L0 thick (n is the number of the lamellae, L0 is lamellar domain spacing) lamellae parallel to the trench wall are formed at each side, while nanotubes are formed away from the trench wall. However, for a trench grafted with PS brushes, nanotubes are formed beside (n+0.25)L0 thick lamellar layers. By adjusting the trench width (W) and the affinity between the block and the wall, various dual nanopatterns consisting of lines and nanotubes are fabricated. Moreover, when the trench wall is selectively deposited by Au, asymmetric dual nanopattern is formed, where different numbers of lines exist on each side wall, while nanotubes are formed in the middle of the trench. The observed morphologies depending on the commensurability condition between W and L0 are consistent with predictions by self-consistent field theory.

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.

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.

Nanopatterns with a Square Symmetry from an Orthogonal Lamellar Assembly of Block Copolymers.

A nanosquare array is an indispensable element for the integrated circuit design of electronic devices. Block copolymer (BCP) lithography, a promising bottom-up approach for sub-10 nm patterning, has revealed a generic difficulty in the production of square symmetry because of the thermodynamically favored hexagonal packing of self-assembled sphere or cylinder arrays in thin-film geometry. Here, we demonstrate a simple route to square arrays via the orthogonal self-assembly of two lamellar layers on topographically patterned substrates. While bottom lamellar layers within a topographic trench are aligned parallel to the sidewalls, top layers above the trench are perpendicularly oriented to relieve the interfacial energy between grain boundaries. The size and period of the square symmetry are readily controllable with the molecular weight of BCPs. Moreover, such an orthogonal self-assembly can be applied to the formation of complex nanopatterns for advanced applications, including metal nanodot square arrays.

Interactions between brush-grafted nanoparticles within chemically identical homopolymers: the effect of brush polydispersity.

We systematically examined the polymer-mediated interparticle interactions between polymer-grafted nanoparticles (NPs) within chemically identical homopolymer matrices through experimental and computational efforts. In experiments, we prepared thermally stable gold NPs grafted with polystyrene (PS) or poly(methyl methacrylate) (PMMA), and they were mixed with corresponding homopolymers. The nanocomposites are well dispersed when the molecular weight ratio of free to grafted polymers, α, is small. For α above 10, NPs are partially aggregated or clumped within the polymer matrix. Such aggregation of NPs at large α has been understood as an autophobic dewetting behavior of free homopolymers on brushes. In order to theoretically investigate this phenomenon, we calculated two particle interaction using self-consistent field theory (SCFT) with our newly developed numerical scheme, adopting two-dimensional finite volume method (FVM) and multi-coordinate-system (MCS) scheme which makes use of the reflection symmetry between the two NPs. By calculating the polymer density profile and interparticle potential, we identified the effects of several parameters such as brush thickness, particle radius, α, brush chain polydispersity, and chain end mobility. It was found that increasing α is the most efficient method for promoting autophobic dewetting phenomenon, and the attraction keeps increasing up to α = 20. At small α values, high polydispersity in brush may completely nullify the autophobic dewetting, while at intermediate α values, its effect is still significant in that the interparticle attractions are heavily reduced. Our calculation also revealed that the grafting type is not a significant factor affecting the NP aggregation behavior. The simulation result qualitatively agrees with the dispersion/aggregation transition of NPs found in our experiments.

Finite volume method for self-consistent field theory of polymers: Material conservation and application.

For the purpose of checking material conservation of various numerical algorithms used in the self-consistent-field theory (SCFT) of polymeric systems, we develop an algebraic method using matrix and bra-ket notation, which traces the Hermiticity of the product of the volume and evolution matrices. Algebraic tests for material conservation reveal that the popular pseudospectral method in the Cartesian grid conserves material perfectly, while the finite-volume method (FVM) is the proper tool when real-space SCFT with the Crank-Nicolson method is adopted in orthogonal coordinate systems. We also find that alternating direction implicit methods combined with the FVM exhibit small mass errors in the SCFT calculation. By introducing fractional cells in the FVM formulation, accurate SCFT calculations are performed for systems with irregular geometries and the results are consistent with previous experimental and theoretical works.

Directed self-assembly of cylinder-forming diblock copolymers on sparse chemical patterns.

Using both theory and experiment, we investigate the possibility of creating perfectly ordered block copolymer nanostructures on sparsely patterned substrates. Our study focuses on scrutinizing the appropriate pattern conditions to avoid undesired morphologies or defects when depositing cylinder-forming AB diblock copolymer thin films on the substrates which are mostly neutral with periodic stripe regions preferring the minority domain. By systematically exploring the parameter space using self-consistent field theory (SCFT), the optimal conditions for target phases are determined, and the effects of the chemical pattern period and the block copolymer film thickness on the target phase stability are also studied. Furthermore, as a sample experimental system, almost perfectly aligned polystyrene-block-poly(methyl methacrylate) (PS-b-PMMA) diblock copolymers are demonstrated. After the pattern transfer process, highly ordered Al nanodot arrays following the initial vertically aligned cylinder pattern are created. This systematic study demonstrates the ability to control the structure and the position of nanopatterns on sparse chemical patterns.

Anomalous rapid defect annihilation in self-assembled nanopatterns by defect melting.

Molecular self-assembly commonly suffers from dense structural defect formation. Spontaneous defect annihilation in block copolymer (BCP) self-assembly is particularly retarded due to significant energy barrier for polymer chain diffusion and structural reorganization. Here we present localized defect melting induced by blending short neutral random copolymer chain as an unusual method to promote the defect annihilation in BCP self-assembled nanopatterns. Chemically neutral short random copolymer chains blended with BCPs are specifically localized and induce local disordered states at structural defect sites in the self-assembled nanopatterns. Such localized "defect melting" relieves the energy penalty for polymer diffusion and morphology reorganization such that spontaneous defect annihilation by mutual coupling is anomalously accelerated upon thermal annealing. Interestingly, neutral random copolymer chain blending also causes morphology-healing self-assembly behavior that can generate large-area highly ordered 10 nm scale nanopattern even upon poorly defined defective prepatterns. Underlying mechanisms of the unusual experimental findings are thoroughly investigated by three-dimensional self-consistent field theory calculation.

Multicomponent nanopatterns by directed block copolymer self-assembly.

Complex nanopatterns integrating diverse nanocomponents are crucial requirements for advanced photonics and electronics. Currently, such multicomponent nanopatterns are principally created by colloidal nanoparticle assembly, where large-area processing of highly ordered nanostructures raises significant challenge. We present multicomponent nanopatterns enabled by block copolymer (BCP) self-assembly, which offers device oriented sub-10-nm scale nanopatterns with arbitrary large-area scalability. In this approach, BCP nanopatterns direct the nanoscale lateral ordering of the overlaid second level BCP nanopatterns to create the superimposed multicomponent nanopatterns incorporating nanowires and nanodots. This approach introduces diverse chemical composition of metallic elements including Au, Pt, Fe, Pd, and Co into sub-10-nm scale nanopatterns. As immediate applications of multicomponent nanopatterns, we demonstrate multilevel charge-trap memory device with Pt-Au binary nanodot pattern and synergistic plasmonic properties of Au nanowire-Pt nanodot pattern.

Nanoporous bicontinuous structures via addition of thermally-stable amphiphilic nanoparticles within block copolymer templates.

Herein, we fabricated the bicontinuous structures from nanocomposites of poly(styrene-b-methyl methacrylate) (PS-b-PMMA) block copolymer and the shell-cross-linked, thermally stable gold nanoparticles (Au NPs). The surface property of Au NPs was controlled with ligands containing various compositions of PS and PMMA so that the resulting Au NPs were selective to PS or PMMA block or nonselective (i.e., neutral) to both blocks. The amphiphilic Au NPs were also prepared by coating the surface of Au NPs with equimolar mixtures of PS and PMMA selective ligands. Consequently, it was found that the morphological behaviors of thermally annealed nanocomposites containing amphiphilic Au NPs and PS-b-PMMA were dramatically different from the case of neutral Au NPs that were coated with nonselective ligands. With increasing the amount of amphiphilic Au NPs, a transition from lamellar to bicontinuous structures was observed, whereas the neutral Au NPs were aggregated within the PS-b-PMMA lamellae. Furthermore, the nanoporous bicontinuous thin films were fabricated on the silicon substrates and the morphological behaviors were quantitatively investigated by grazing-incidence small-angle X-ray scattering (GISAXS) analysis.

Graphoepitaxy of block-copolymer self-assembly integrated with single-step ZnO nanoimprinting.

A highly efficient, ultralarge-area nanolithography that integrates block-copolymer lithography with single-step ZnO nanoimprinting is introduced. The UV-assisted imprinting of a photosensitive sol-gel precursor creates large-area ZnO topographic patterns with various pattern shapes in a single-step process. This straightforward approach provides a smooth line edge and high thermal stability of the imprinted ZnO pattern; these properties are greatly advantageous for further graphoepitaxial block-copolymer assembly. According to the ZnO pattern shape and depth, the orientation and lateral ordering of self-assembled cylindrical nanodomains in block-copolymer thin films could be directed in a variety of ways. Significantly, the subtle tunability of ZnO trench depth enabled by nanoimprinting, generated complex hierarchical nanopatterns, where surface-parallel and surface-perpendicular nanocylinder arrays are alternately arranged. The stability of this complex morphology is confirmed by self-consistent field theory (SCFT) calculations. The highly ordered graphoepitaxial nanoscale assembly achieved on transparent semiconducting ZnO substrates offers enormous potential for photonics and optoelectronics.

One-dimensional metal nanowire assembly via block copolymer soft graphoepitaxy.

We accomplished a facile and scalable route to linearly stacked, one-dimensional metal nanowire assembly via soft graphoepitaxy of block copolymers. A one-dimensional nanoscale lamellar stack could be achieved by controlling the block copolymer film thickness self-assembled within the disposable topographic confinement and utilized as a template to generate linear metal nanowire assembly. The mechanism underlying this interesting morhpology evolution was investigated by self-consistent field theory. The optical properties of metal nanowire assembly involved with surface plasmon polariton were investigated by first principle calculations.

Positioning Janus nanoparticles in block copolymer scaffolds.

The periodic domains formed by block copolymer melts have been heralded as potential scaffolds for arranging nanoparticles in 3D space, provided we can control the positioning of the particles. Recent experiments have located particles at the domain interfaces by grafting mixed brushes to their surfaces. Here the underlying mechanism, which involves the transformation into Janus particles, is investigated with self-consistent field theory using a new multi-coordinate-system algorithm.

Morphology selection of nanoparticle dispersions by polymer media.

Designable media can control properties of nanocomposite materials by spatially organizing nanoparticles. Here we theoretically study particle organization by ultrathin polymer films of grafted chains ("brushes"). Polymer-soluble nanoparticles smaller than a brush-determined threshold disperse in the film to a depth scaling inversely with particle volume. In the polymer-insoluble case, aggregation is directed: provided particles are nonwetting at the film surface, the brush stabilizes the dispersion and selects its final morphology of giant elongated aggregates with a brush-selected width.