A surface interaction model for self-assembly of block copolymers under soft confinement.

The surface interaction between substrates and block copolymers is one of the most important factors that control the alignment of self-assembled domains under thin film confinement. Most previous studies simply modeled substrates modified by grafting polymers as a hard wall with a specified surface energy, leading to an incomplete understanding of the role of grafted polymers. In this study, we propose a general model of surface interactions where the role of grafted polymers is decomposed into two independent contributions: the surface preference and the surface softness. Based on this model, we perform a numerical analysis of the stability competition between perpendicular and parallel lamellae of symmetric diblock copolymers on substrates modified by homopolymers using self-consistent field theory. The effects of the surface preference and the surface softness on the alignment of lamellar domains are carefully examined. A phase diagram of the alignment in the plane of the surface preference parameter and the surface softness parameter is constructed, which reveals a considerable parameter window for preparing stable perpendicular lamellae even on highly preferential substrates.

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

We present a fast and accurate numerical method for the self-consistent field theory calculations of confined polymer systems. It introduces an exponential time differencing method (ETDRK4) based on Chebyshev collocation, which exhibits fourth-order accuracy in temporal domain and spectral accuracy in spatial domain, to solve the modified diffusion equations. Similar to the approach proposed by Hur et al. [Macromolecules 45, 2905 (2012)], non-periodic boundary conditions are adopted to model the confining walls with or without preferential interactions with polymer species, avoiding the use of surface field terms and the mask technique in a conventional approach. The performance of ETDRK4 is examined in comparison with the operator splitting methods with either Fourier collocation or Chebyshev collocation. Numerical experiments show that our exponential time differencing method is more efficient than the operator splitting methods in high accuracy calculations. This method has been applied to diblock copolymers confined by two parallel flat surfaces.