Percolation in binary mixtures of linkers and particles: Chaining vs branching

Equilibrium gels of colloidal particles can be realized through the introduction of a second species, a linker that mediates the bonds between colloids. A gel forming binary mixture whose linkers can self-assemble into linear chains while still promoting the aggregation of particles is considered in this work. The particles are patchy particles with f_C patches of type C and the linkers are patchy particles with 2 patches of type A and f_B patches of type B. The bonds between patches of type A (AA bonds) promote the formation of linear chains of linkers. Two different ways (model A and model B) of bonding the linkers to the particles-or inducing branching-are studied. In model A, there is a competition between chaining and branching, since the bonding between linkers and particles takes place through AC bonds only. In model B, the linkers aggregate to particles through bonds BC only, making chaining and branching independent. The percolation behavior of these two models is studied in detail, employing a generalized Flory-Stockmayer theory and Monte Carlo simulations. The self-assembly of linkers into chains reduces the fraction of particles needed for percolation to occur (models A and B) and induces percolation when the fraction of particles is high (model B). Percolation by heating and percolation loops in temperature-composition diagrams are obtained when the formation of chains is energetically favorable by increasing the entropic gain of branching (model A). Chaining and branching are found to follow a model dependent relation at percolation, which shows that, for the same composition, longer chains require less branching for percolation to occur.