Random growth lattice filling model of percolation: a crossover from continuous to discontinuous transition

Abstract

A random growth lattice filling model of percolation with touch and stop growth rule is developed and studied numerically on a two dimensional square lattice. Nucleation centers are continuously added one at a time to the empty sites and the clusters are grown from these nucleation centers with a tunable growth probability g. As the growth probability g is varied from 0 to 1 two distinct regimes are found to occur. For g ≤ 0.5, the model exhibits continuous percolation transitions as ordinary percolation whereas for g ≥ 0.8 the model exhibits discontinuous percolation transitions. The discontinuous transition is characterized by discontinuous jump in the order parameter, compact spanning cluster and absence of power law scaling of cluster size distribution. Instead of a sharp tricritical point, a tricritical region is found to occur for 0.5 < g < 0.8 within which the values of the critical exponents change continuously till the crossover from continuous to discontinuous transition is completed.

Structure Elucidation of 2D Polymer Monolayers Based on Crystallization Estimates Derived from Tip-Enhanced Raman Spectroscopy (TERS) Polymerization Conversion Data

“…Thus, the monolayer covalent sheets reported here are 2D polymers and contain fully ordered areas amounting to 710 nm 2 for the case with 93% conversion, separated by areas amounting to approximately 2.8% of the total sheet area likely to contain unreacted anthracene pairs as the defects. Random growth models have been studied in the past mostly to address percolation behavior in networks , and can be adapted to monomers with various growth multiplicities. In the present case, this leads to a simple relationship between X and C and the size distribution of connected, defect-free crystalline regions.…”