Density and concentration field description of nonperiodic structures

Abstract

We propose a simple nonlocal energy functional that is suitable for the continuum field characterization of nonperiodic and localized textures. The phenomenological functional is based on the pairwise direction-dependent interaction of field gradients that are separated by a fixed distance. In an appendix, we describe the numerical minimization of our functional. On that basis, we investigate the kinetic evolution of threadlike stripe patterns that are created by the functional when we start from an initially disordered state. At later stages, we find a coarse graining that shows the same scaling behavior as was obtained for the Cahn-Hilliard equation. In fact, the Cahn-Hilliard model is contained in our characterization as a limiting case. A slight modification of our model omits this coarse graining and leads to nonperiodic stripe phases. For the latter case, we investigate the temporal evolution of the defects (end points) of the threadlike stripes. In view of actual applications of this functional, we discuss the characterization of processes observed for polymeric systems and vesicles. The statistics of the growth of the threadlike structures is compared to the case of step-growth polymerization reactions. Furthermore, we demonstrate that the functional may be applied for the study of vesicles in a continuum field description. Basic features, such as the tendency of tank treading in simple shear flows and parachute folding in pipe flows, are reproduced.