Percolation phase diagrams for multi-phase models built on the overlapping sphere model

The overlapping sphere (OS) percolation model gives a two-phase microstructure (matrix plus inclusions) that is useful for testing composite material ideas and other applications, as well as serving as a paradigm of overlapping object percolation and phase transitions. Real materials often have more than two phases, so it is of interest to extend the applicability of the OS model. A flexible variant of the OS model can be constructed by randomly assigning the spheres different phase labels, according to a uniform probability distribution, as they are inserted one by one into the matrix. The resulting three or more phase models can have different amounts of percolating and non-percolating phases, depending on the volume fraction of each phase and the total OS volume fraction. A three-dimensional digital image approach is used to approximately map out the percolation phase diagram of such models, explicitly up to four phases (one matrix plus three spherical inclusion phases) and implicitly for N>4 phases. For the three phase model, it was found that a single OS sub-phase has a percolation threshold that ranges from about a volume fraction of 0.16, when the matrix volume fraction is about 0.01, to about 0.30, at a matrix volume fraction of about 0.7. The approximate analytical dependence of this sub-phase percolation threshold on the defining model parameters serves to guide the building of the percolation phase diagram for the N-phase model, and is used to determine the maximum value of N(N=6) at which all N phases can be simultaneously percolated.