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.