Spectral collocation methods for polymer brushes

Abstract

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.

Stimuli-Responsive Brushes with Active Minority Components: Monte Carlo Study and Analytical Theory

“…In the simulation community, a variety of numerical methods have been developed and used to investigate polymer brush systems, ranging from molecular dynamics [23], dissipative particle dynamics [24], and Monte Carlo (MC) simulations [25][26][27][28][29] to numerical selfconsistent field (SCF) calculations [30][31][32][33][34][35]. The present study mainly relies on MC simulations, while for comparison we also present results obtained from 3-d SCF theory.…”

“…Similar problems were reported in Refs. [33,34]. Therefore, we adopt the Doublas-Brian scheme [68], a realspace finite-difference method which did not suffer from this problem.…”

Exponential time differencing methods with Chebyshev collocation for polymers confined by interacting surfaces

“…For the sake of comparison, the operator splitting methods on equispaced grid presented in Ref. 19 and the operator splitting methods on CGL grid (OSCHEB) proposed by Hur et al 22 are also implemented. Note that the operator splitting methods on equispaced grid are only applicable to DBC and NBC.…”

“…μ and α are two constants which control the width and asymptotic angle of the hyperbola. Applying the trapezoidal rule to evaluate the integral, we have (19) where 2M + 1 is the number of equally spaced points on the contour and their spacing is = 1.0818/M, and u k = k and v k = v(u k ). According to Weideman and Trefethen, 31 we choose μ = 4.4921M/h and α = 1.1721 which optimize the accuracy of Eq.…”

“…18 Fortunately, MDEs with above two kinds of boundary conditions, if they are homogeneous, can still be solved by the pseudospectral method, provided that the standard Fourier basis (for PBC) is replaced by the sine (for DBC) or cosine (for NBC). 19 However, such Fourier sine (cosine) methods often lose spectral accuracy, i.e., the exponential convergence, since it requires the solution of the MDE is infinite smooth and its all even (odd) derivatives of second and higher order must vanish at the boundary. But these conditions are hardly fulfilled in confined polymer systems.…”