Periodicity and global order parameter of hexagonally packed cylinders in a periodic box

In all molecular simulations of periodic ordered morphologies (such as those formed by block copolymers), the periodic boundary conditions (PBCs) of the simulation box usually do not match the bulk periodicity L₀ of the morphology, thus changing the structure and even the stability of the morphologies obtained in the simulations. To address this problem for hexagonally packed cylinders, we first proposed a general method of calculating the periodicity of such cylinders in a cuboid simulation box with the PBCs applied in all directions, which further allows one to enumerate all possible orientations and periodicities of such cylinders within an estimated range that can fit into a cuboid box of given lengths. We then showed how to choose the lengths of a cuboid box such that regular-hexagonally packed (RHP) cylinders with given intercylinder distance and orientation can fit into the box. Next, taking as an example the dissipative particle dynamics (DPD) simulations of a cylinder-forming diblock copolymer melt, we showed that L₀ of RHP cylinders oriented along the body diagonal of a cubic box is found when all the off-diagonal elements of the pressure tensor vanish. Finally, based on our general method of calculating the periodicity of hexagonally packed cylinders, we designed a global order parameter for such cylinders, which takes into account their ordering only for the orientations that can fit into the simulation box. Using again the DPD simulations, we showed that our global order parameter can be used to quantify the formation of hexagonally packed cylinders in each collected configuration and to monitor their orientation (thus periodicity) during the simulation run.