Percolation of colloids with distinct interaction sites

We generalize the Flory-Stockmayer theory of percolation to a model of associating (patchy) colloids, which consists of hard spherical particles, having on their surfaces $f$ short-ranged-attractive sites of $m$ different types. These sites can form bonds between particles and thus promote self-assembly. It is shown that the percolation threshold is given in terms of the eigenvalues of a $m\times m$ matrix, which describes the recursive relations for the number of bonded particles on the $i\text{th}$ level of a cluster with no loops; percolation occurs when the largest of these eigenvalues equals unity. Expressions for the probability that a particle is not bonded to the giant cluster, for the average cluster size and the average size of a cluster to which a randomly chosen particle belongs, are also derived. Explicit results for these quantities are computed for the case $f=3$ and $m=2$. We show how these structural properties are related to the thermodynamics of the associating system by regarding bond formation as a (equilibrium) chemical reaction. This solution of the percolation problem, combined with Wertheim’s thermodynamic first-order perturbation theory, allows the investigation of the interplay between phase behavior and cluster formation for general models of patchy colloids.

Figure 1

Left: schematic representation of the particles—hard spheres (large disks) with bonding sites on their surfaces (small disks); there are $f$ bonding sites of $m$ types ($f_{\alpha }$ of type $\alpha$); here $f=5$, $m=3$, $f_1=3$, and $f_2=f_3=1$. Right: when two bonding sites ($\alpha$ and $\beta$) overlap, a bond $\alpha \beta$ forms, decreasing the energy by ${ϵ}_{\alpha \beta }$; here a bond 13 has formed.

Figure 2

Schematic representation of the treelike clusters. After choosing a random particle (at the top of the figure), the cluster can be represented by levels as shown; on each level there are $n_{\alpha }$ bonded sites of type $\alpha$. The arrows represent the bonds that, in this description, allow one to go from one level to the next; each bond (arrow) connecting a site $\alpha$ to a site $\beta$ occurs with probability $p_{\alpha \to \beta }$.

Figure 3

(Color online) Two realizations of the model $2A+1B$: solid and dashed (black) lines are for ${ϵ}_{BB}=0,{ϵ}_{AB}=0.75{ϵ}_{AA}$; dashed-dotted and dotted (red) lines are for ${ϵ}_{AB}=0,{ϵ}_{BB}=0.75{ϵ}_{AA}$. (a) Phase diagrams and percolation lines; (b) eigenvalues of $\tilde{T}$ (${\lambda }_{+}$: solid and dot-dashed lines; ${\lambda }_{-}$: dashed and dotted lines); (c) $Q_A^2{Q}_B$, fraction of particles in the sol phase; (d) $N_w$ (solid and dot-dashed lines) and $N_n$ (dashed and dotted lines). In (b)–(d) all quantities are calculated as functions of temperature at the critical density.