Nonequilibrium structure in sequential assembly

The assembly of monomeric constituents into molecular superstructures through sequential-arrival processes has been simulated and theoretically characterized. When the energetic interactions allow for complete overlap of the particles, the model is equivalent to that of the sequential absorption of soft particles on a surface. In the present work, we consider more general cases by including arbitrary aggregating geometries and varying prescriptions of the connectivity network. The resulting theory accounts for the evolution and final-state configurations through a system of equations governing structural generation. We find that particle geometries differ significantly from those in equilibrium. In particular, variations of structural rigidity and morphology tune particle energetics and result in significant variation in the nonequilibrium distributions of the assembly in comparison to the corresponding equilibrium case.

Figure 1

The probability of site occupation $\varphi$ as a function of $\delta$ at various density values $\mathrm{\Phi }$ for three interaction networks: (a) complete $\mathcal{K}$, (b) tethered $\mathcal{T}$, and (c) ring $\mathcal{R}$. The blue circular markers, orange dashed curves, and red solid curves correspond to the simulation, the first-order approximation in Eq. (14), and the second-order approximation in Eq. (16), respectively. The lower cyan solid curves in the top panels correspond to the equilibrium values of $\varphi$ as measured from simulation. The insets in each panel are graphs of corresponding networks for $n=\left\{3,4,5\right\}$ particles. (d)–(f) The probability ${\varphi }_n$ for observing $n$ particles in a site as a function of $\delta$ for the corresponding network (shown above), as measured from simulation.

Figure 2

(a) The probability of site occupation $\varphi$ and (b) growth $d\varphi /d\mathrm{\Phi }$ as a function of density $\mathrm{\Phi }$ for various values of $\delta$. The energetics correspond to the complete interaction case $\mathcal{K}$. The inset shows the growth rate on a log scale. The solid curves are the results measured from simulation, and the dashed curves are the results given by Eq. (16).