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



Edward J. Garboczi


The overlapping sphere (OS) model gives a two-phase microstructure (matrix plus inclusions) that is useful for testing out composite material ideas and other applications, as well as serving as a paradigm of continuum percolation and phase transitions. Real materials often have more than two phases, so it is of interest to extend 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 into the matrix. The resulting three or more phase models can have a different makeup of percolating and non-percolating phases, depending on the volume fractions of each phase and the total OS volume fraction. A 3D 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. 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 0.29, when the total spherical inclusion phase just percolates at a matrix volume fraction of 0.71. The number of sphere-sphere bonds per sphere for this sub-phase at the percolation threshold also varies with the total volume fraction of spheres. The approximate analytical dependence of this sub-phase percolation threshold on the defining 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.
Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)


percolation, spheres, overlapping, phase diagram, digital image, continuum percolation, threshold, excluded volume, bonds


Garboczi, E. (2015), Percolation phase diagrams for multi-phase models built on the overlapping sphere model, Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), [online], (Accessed June 13, 2024)


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Created September 16, 2015, Updated November 10, 2018