Periodicity and Order Parameter of Hexagonally Packed Cylindrical Phase in a Periodic Box

In molecular simulations of self-assembled periodic structures (e.g., those formed by block copolymers), the periodic boundary conditions (PBCs) of the simulation box limit the periodicity (thus the orientation) of the periodic structures to discrete values that are commensurate with the periodicity of the box. For the most commonly used orthorhombic simulation boxes, while the case of cubic phases (e.g., the body-centered cubic spheres or the double gyroid) is straightforward to analyze and one of us proposed a general formula to calculate the lamellar period, the periodicity of hexagonally packed cylindrical phase has only been analyzed by one of us in a few special cases (Macromolecules 34, 3458 (2001); J. Chem. Phys. 138, 194904 (2013)). Here we propose a general method for calculating the periodicity of the cylindrical phase in an orthorhombic box. Based on this, we further propose an order parameter of the cylindrical phase suitable for the study of phase transitions in molecular simulations.