Rigid-floppy percolation threshold

The mathematical limits for the percolation of rigid and floppy clusters throughout continuous networks are considered. Results are compared to the previous prediction of Thorpe based on the assumption that the threshold condition is the equality of degrees of freedom and the number of constraints. It is demonstrated that, in two-dimensional systems, the thresholds values predicted here and in the Thorpe model are relatively close. Our investigation demonstrates that, in 3D, there is a range of strong bond concentrations in which both floppy and rigid clusters percolate simultaneously. Depending on the particular property, either the rigid or the floppy cluster plays the key role. This explains the existence of solid electrolytes. They are solid because of the percolating rigid skeleton and at the same time have high conductivity controlled by the floppy cluster that also percolates.