Finding the bulk periodicity of lamellar and cylindrical structures using the pressure tensor

It has been known for nearly thirty years that in all molecular simulations of periodic ordered morphologies (such as those formed by block copolymers), when the periodic boundary conditions of the simulation box do not match the bulk periodicity L₀ of the morphology, they change the structure and even the stability of the morphologies obtained in the simulations. Few studies, however, have focused on finding L₀ in simulations. Taking dissipative particle dynamics simulations of asymmetric diblock copolymer melts as an example, we found a simple way of using the pressure tensor, which can be readily calculated in molecular simulations in the continuum, to find L₀ of lamellae and (regular-hexagonally packed) cylinders regardless of their orientation in (cuboid) simulation boxes. Variation of the diagonal and off-diagonal elements of the pressure tensor with the orientation of lamellae and cylinders in the box is explained by coordinate system rotation and confirmed by our simulation results. We also showed that the pressure tensor cannot be used to find L₀ of 3D periodic ordered structures with a cubic symmetry such as the double gyroid.