Recently there has been a strong interest in the area of defect formation in ordered structures on curved surfaces. Here we explore the closely related topic of self-assembly in thin block copolymer melt films confined to the surface of a sphere. Our study is based on a self-consistent field theory (SCFT) model of block copolymers that is numerically simulated by spectral collocation with a spherical harmonic basis and an extension of the Rasmussen-Kalosakas operator splitting algorithm [J. Polym. Sci. Part B: Polym. Phys. 40, 1777 (2002)]. In this model, we assume that the composition of the thin block copolymer film varies only in longitude and colatitude and is constant in the radial direction. Using this approach we are able to study the formation of defects in the lamellar and cylindrical phases, and their dependence on sphere radius. Specifically, we compute ground-state (i.e., lowest-energy) configurations on the sphere for both the cylindrical and lamellar phases. Grain boundary scars are also observed in our simulations of the cylindrical phase when the sphere radius surpasses a threshold value R_{c} approximately 5d , where d is the natural lattice spacing of the cylindrical phase, which is consistent with theoretical predictions [Bowick, Phys. Rev. B 62, 8738 (2000); Bausch, Science 299, 1716 (2003)]. A strong segregation limit approximate free energy is also presented, along with simple microdomain packing arguments, to shed light on the observed SCFT simulation results.

An efficient numerical self-consistent field theory (SCFT) algorithm is developed for treating structured polymers on spherical surfaces. The method solves the diffusion equations of SCFT with a pseudo-spectral approach that combines a spherical-harmonics expansion for the angular coordinates with a modified real-space Crank–Nicolson method for the radial direction. The self-consistent field equations are solved with Anderson-mixing iterations using dynamical parameters and an alignment procedure to prevent angular drift of the solution. A demonstration of the algorithm is provided for thin films of diblock copolymer grafted to the surface of a spherical core, in which the sequence of equilibrium morphologies is predicted as a function of diblock composition. The study reveals an array of interesting behaviors as the block copolymer pattern is forced to adapt to the finite surface area of the sphere

Cellulose nanofibrils (CNFs) are a class of cellulosic nanomaterials with high aspect ratios that can be extracted from various natural sources. Their highly crystalline structures provide the nanofibrils with excellent mechanical and thermal properties. The main challenges of CNFs in nanocomposite applications are associated with their high hydrophilicity, which makes CNFs incompatible with hydrophobic polymers. In this study, highly transparent and toughened poly(methyl methacrylate) (PMMA) nanocomposite films were prepared using various percentages of CNFs covered with surface carboxylic acid groups (CNF-COOH). The surface groups make the CNFs interfacial interaction with PMMA favorable, which facilitate the homogeneous dispersion of the hydrophilic nanofibrils in the hydrophobic polymer and the formation of a percolated network of nanofibrils. The controlled dispersion results in high transparency of the nanocomposites. Mechanical analysis of the resulting films demonstrated that a low percentage loading of CNF-COOH worked as effective reinforcing agents, yielding more ductile and therefore tougher films than the neat PMMA film. Toughening mechanisms were investigated through coarse-grained simulations, where the results demonstrated that a favorable polymer-nanofibril interface together with percolation of the nanofibrils, both facilitated through hydrogen bonding interactions, contributed to the toughness improvement in these nanocomposites.

We provide an in-depth study of pseudo-spectral numerical methods associated with modeling the self-assembly of molten mixed polymer brushes in the framework of self-consistent field theory (SCFT). SCFT of molten polymer brushes has proved numerically challenging in the past because of sharp features that arise in the self-consistent pressure field at the grafting surface due to the chain end tethering constraint. We show that this pressure anomaly can be reduced by smearing the grafting points over a narrow zone normal to the surface in an incompressible model, and/or by switching to a compressible model for the molten brush. In both cases, we use results obtained from a source (delta function) distribution of grafting points as a reference. At the grafting surface, we consider both Neumann and Dirichlet conditions, where the latter is paired with a masking method to mimic a confining surface. When only the density profiles and relative free energies of two comparison phases are of interest, either source or smeared distributions of grafting points can be used, but a smeared distribution of grafting points exhibits faster convergence with respect to the number of chain contour steps. Absolute free energies converge only within the smeared model. In addition, when a sine basis is used with the masking method and a smeared distribution, fewer iterations are necessary to converge the SCFT fields for the compressible model. The numerical methods described here and investigated in one-dimension will provide an enabling platform for computationally more demanding three-dimensional SCFT studies of a broad range of mixed polymer brush systems.