Growth transitions and critical behavior in the non-equilibrium aggregation of short, patchy nanorods

Abstract

We have carried out Monte Carlo simulations to study the non-equilibrium aggregation of short patchy nanorods in two dimensions. Below a critical value of patch size ($p_c = 0.31675$), the aggregates have finite sizes with small radii of gyration, $R_g$. At $p_c$, the average radius of gyration shows a power law increase with time such that $<R_g>\sim t^{\gamma }$, where $\gamma =0.411 \pm 0.006$. Above, $p_c$, the aggregates are fractal in nature and their fractal dimension depends on the value of patch size. These morphological differences are due to the fact that below the critical value of patch size ($p_{c}$), the growth of the clusters is suppressed and the system reaches an ‘absorbed state.’ Above $p_{c}$, the system reaches an ‘active state,’ in which the cluster size keeps growing with a fixed rate at long times. Thus, the system encounters a non-equilibrium phase transition. Close to the transition, the growth rate scales as $\varGamma (t) \sim t^{-\alpha }$, where $\alpha = 0.160 \pm 0.030$. The long-time growth rate varies as $\varGamma (\infty )\sim (p-p_c)^\beta$ where $\beta = 0.279 \pm 0.034$. These scaling exponents indicate that the transition belongs to the directed percolation universality class. The patchy nanorods also display a threshold patch size ($p_{t}$), beyond which the long-time growth rate remains constant. We present geometric arguments for the existence of $p_t$. The fractal dimension of the aggregates increases from 1.75, at $p_c$, to 1.81, at $p_t$. It remains constant beyond $p_t$.

Graphic abstract