Spectral collocation methods for polymer brushes

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.