Linear Response and Stability of Ordered Phases of Block Copolymer Melts

Abstract

An efficient pseudo-spectral numerical method is introduced for calculating a self-consistent field (SCF) approximation for the linear susceptibility of ordered phases in block copolymer melts (sometimes referred to as the random phase approximation). Our method is significantly more efficient than that used in the first calculations of this quantity by Shi, Laradji and coworkers, allowing for the study of more strongly segregated structures. We have re-examined the stability of several phases of diblock copolymer melts, and find that some conclusions of Laradji et al. regarding the stability of the Gyroid phase were the result of insufficient spatial resolution. We find that an epitaxial (k = 0) instability of the Gyroid phase with respect to the hexagonal phase that was considered previously by Matsen competes extremely closely with an instability that occurs at a nonzero crystal wavevector k.

Nucleation of Ordered Phases in Block Copolymers

“…The boundary between different ordered phases can be specified by, e.g., È s ¼ 0:5. The nucleation of cylinders from a metastable lamellar phase is examined at a fixed f ¼ 0:45 and varying N in the range between the mean-field spinodal point of the lamellae and the L-C coexistence point [5,6]. The initial cylinders were oriented parallel to the lamellar plane, in agreement with previous studies [9].…”

“…When the ordered phase is thermodynamically unstable, phase transition proceeds via spinodal decomposition. For block copolymer phases undergoing spinodal decomposition, theoretical studies have been carried to understand the most probable kinetic pathways by examining the most unstable modes of the ordered phases [4][5][6][7]. When the ordered phase is metastable, phase transition proceeds via nucleation and growth [8].…”

“…The equilibrium phase behavior of a diblock copolymer melt is obtained by finding the minima of the free energy functional or solutions of the Euler-Lagrange equation H w AE ¼ 0. This is a highly nontrivial problem, but by now, we have a wealth of information regarding the equilibrium phase diagram [3,15] including the linear stability [5,6] of the different phases. On the other hand, nucleation of an ordered phase from a metastable phase in block copolymers has received less attention, with the exception of the work of Wickham et al [9] and the work of Matsen for the epitaxial transitions of C $ S and C $ G [16].…”

Living Polymer Systems at a Solid Substrate: Computer Simulation of a Soft, Coarse-Grained Model and Self-Consistent Field Theory

“…The one-dimensional modified diffusion equations for the chain propagators are solved by a combination of the pseudospectral method 20,21 and Richardson extrapolation 22 under Dirichlet boundary conditions. 16 We provide further details of these calculations in Appendix A.…”

High-order sampling schemes for path integrals and Gaussian chain simulations of polymers

“…Fully spectral techniques do not involve any discretization error of the Gaussian chain contour. 22 Fourth-order accuracy in solving this partial differential equation can be achieved by combining a second-order pseudospectral algorithm with Richardson extrapolation 23 or by treating the Laplacian operator implicitly with a fourth-order backward differentiation formula and the source term explicitly via a fourth-order Adams-Bashford scheme. 24 The numerical performance of these algorithms has been carefully examined in the literature.…”