Rigid m-percolation in limited-valence gels

Determining the onset of rigidity in gels is a fundamental challenge with significant practical implications across different applications. Limited-valence, or patchy-particle, systems have proven to be a valuable model to study the relationship between microscopic interactions and macroscopic mechanical properties. It has been suggested that the emergence of rigidity coincides with the formation of an infinitely spanning cluster of particles with at least three bonds. This work explores this hypothesis, its implications, and its broader applicability across a range of system parameters, by associating the emergence of rigidity with m-percolation transition for m=3. The properties of m-percolation are developed using a mean-field theoretical approach validated with numerical simulations, and used to build phase and rigidity diagrams for different particle valences of both single-component systems and binary mixtures of patchy particles. The difference between connectivity and rigidity percolation thresholds is found to reduce with increasing valence, providing an explanation for the challenges encountered in experimental attempts to distinguish isotropic connectivity percolation from the onset of rigidity. For binary mixtures, we found a robust minimum average valence, below which the gel is never rigid.