Percolation in binary mixtures of linkers and particles: chaining {\it {vs}} branching

TL;DR

This study investigates how the formation of linker chains versus branching affects percolation in binary colloidal mixtures, revealing that chain self-assembly influences gel formation and percolation thresholds.

Contribution

It introduces two models of linker-particle bonding, analyzing their percolation behavior with theoretical and simulation methods, highlighting the role of chain formation.

Findings

Self-assembly of linkers into chains lowers percolation threshold.

Percolation can occur at high particle fractions in model B.

Chaining reduces the amount of branching needed for percolation.

Abstract

Equilibrium gels of colloidal particles can be realized through the introduction of a second species, a linker that mediates the bonds between the 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 is done through $AC$ bonds only. In model B linkers aggregate to particles through bonds $BC$ only, making chaining and branching independent. The percolation behaviour 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 favourable, 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.