Gelation threshold of cross-linked polymer brushes

The cross-linking of polymer brushes is studied using the bond-fluctuation model. By mapping the cross-linking process into a two-dimensional (2D) percolation problem within the lattice of grafting points, we investigate the gelation transition in detail. We show that the particular properties of cross-linked polymer brushes can be reduced to the distribution of bonds which are formed between the grafted chains, and we propose scaling arguments to relate the gelation threshold to the chain length and the grafting density. The gelation threshold is lower than the percolation threshold for 2D bond percolation because of the longer range and broad distribution of bonds formed by the cross-linking process. We term this type of percolation problem star percolation. We observe a broad crossover from mean-field to critical percolation behavior by analyzing the cluster size distribution near the gelation threshold.