Smoluchowski equations for linker-mediated irreversible aggregation†
We developed a generalized Smoluchowski framework to study linker-mediated aggregation, where linkers and particles are explicitly taken into account. We assume that the bonds between linkers and particles are irreversible, and that clustering occurs through limited diffusion aggregation. The kernel is chosen by analogy with single-component diffusive aggregation but the clusters are distinguished by their number of particles and linkers. We found that the dynamics depends on three relevant factors, all tunable experimentally: (i) the ratio of the diffusion coefficients of particles and linkers; (ii) the relative number of particles and linkers; and (iii) the maximum number of linkers that may bond to a single particle. To solve the Smoluchoski equations analytically we employ a scaling hypothesis that renders the fraction of bondable sites of a cluster independent of the size of the cluster, at each instant. We perform numerical simulations of the corresponding lattice model to test this hypothesis. We obtain results for the asymptotic limit, and the time evolution of the bonding probabilities and the size distribution of the clusters. These findings are in agreement with experimental results reported in the literature and shed light on unexplained experimental observations.