An algorithm for detecting percolating structures in periodic systems – Application to polymer networks

We consider the problem of detecting a percolating structure in an off-lattice model polymer system when periodic boundary conditions are used. Physically, with increasing polymer density, the point at which this first occurs is the gel point. A connected structure spans all space and the system becomes solid-like. This problem is similar to that of finding connected paths in lattice bond percolation and site percolation models. We show that algorithms that detect these structures in finite systems will not always yield the correct answer for a system that is periodically repeated in space (uses periodic boundary conditions). Here we describe an algorithm that detects clusters that connect to themselves over an arbitrary number of periodic replicas. Because of the periodic replication this means that they are connected over all space. For system sizes that are typically tractable in simulations we find a relatively minor proportion of configurations are mis-classified for a lattice model. However, the fraction that is mis-classified is significant for the polymer system. Mis-classified configurations when included in the calculation of ensemble averages will include configurations with spurious physical properties. Our algorithm allows this to be corrected for.