Density and concentration field description of nonperiodic structures

TL;DR

This paper introduces a nonlocal energy functional for modeling nonperiodic, localized textures, demonstrating its application in simulating pattern evolution, defect dynamics, and potential uses in polymeric and vesicle systems.

Contribution

It presents a novel nonlocal functional that captures nonperiodic structures and includes a numerical approach, extending the Cahn-Hilliard model and enabling applications to complex materials.

Findings

Functional reproduces Cahn-Hilliard scaling behavior

Modification yields nonperiodic stripe phases

Functional applicable to polymeric systems and vesicles

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 thread-like 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 thread-like stripes. In view of possible applications of this functional, we discuss a possible characterization of polymeric systems and vesicles. The statistics of the growth of the thread-like 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.