Since we are interested in the thermodynamics of mixing different species, we only consider cluster diagrams where the entirety of one species are fully-connected. In other words, all intra-species couplings are fully turned on via Eq. 1. We therefore have no need to represent intra-species coupling in our diagrams (since they are always fully-connected), and represent a given species with a double circle (the double circle represents the internal structure within the phase). For example, an isolated water phase with configurational integral, Z W , is represented as: