(167m) Periodicity of Lamellar and Hexagonally Packed Cylindrical Phases in a Periodic Box

In all molecular simulations of periodic ordered morphologies (POMs, such as those formed by block copolymers), the periodic boundary conditions (PBCs) limit the periodicity L of POMs to discrete values that must be commensurate with the periodicity imposed by the PBCs. For the commonly used cuboid simulation boxes, although the case of cubic phases (e.g., the body-centered cubic spheres or the double gyroid) is straightforward to analyze, those of lamellae and cylinders are complicated by the various orientations they can have in the box. While one of us proposed a general formula to calculate L of lamellae (J. Chem. Phys. 2000, 112, 450), that of cylinders has rarely been calculated. More importantly, when L is different from the bulk periodicity L₀ of POMs, the PBCs change the structure and even the stability of POMs obtained in the simulations.

Here we first propose a general method for calculating the periodicity of hexagonally packed cylinders in a cuboid box. Based on this, we further propose a global order parameter of the cylindrical phase suitable for the study of phase transitions in molecular simulations. We also show how to choose the lengths of a cuboid box such that regular-hexagonally packed (RHP) cylinders with given L and orientation can fit into the box. Finally, we show how to use pressure tensor, which can be readily calculated in off-lattice molecular simulations, to determine L₀ of both lamellae and RHP cylinders regardless of their orientation.