Clustering of low-valence particles: Structure and kinetics

We compute the structure and kinetics of two systems of low-valence particles with three or six freely oriented bonds in two dimensions. The structure of clusters formed by trivalent particles is complex with loops and holes, while hexavalent particles self-organize into regular and compact structures. We identify the elementary structures which compose the clusters of trivalent particles. At initial stages of clustering, the clusters of trivalent particles grow with a power-law time dependence. Yet at longer times fusion and fission of clusters equilibrates and clusters form a heterogeneous phase with polydispersed sizes. These results emphasize the role of valence in the kinetics and stability of finite-size clusters.

Figure 1

Snapshot of simulations of $3V$ and $6V$ particles at 0.3 s after the onset of simulation. Initially particles are placed at random positions. They perform Brownian motion and interact at short distances. Interaction energy is $8.2kT$. $N=3000$. $\text{Concentration}=0.1$.

Figure 2

Clusters of 300 $6V$ or $3V$ particles (a) were analyzed to determine the radial correlation function (b). The color of particles represents the current number of bonds a particle has formed. Light blue (light gray): 6, cyan (light gray): 5, blue (gray): 4, dark blue (dark gray): 3, black: 2, red (gray): 1, yellow (very light gray): 0. (b) Correlation function. The potential well $\varepsilon$ is 3, 6, and $8kT$ for gray, light gray, and black lines, respectively.

Figure 3

Internal cluster structures. Magnified view of the particle organization in a cluster of 500 particles. Interaction energy is $7kT$. (a) $6V$ free bond model. (b) $3V$ fixed bond model. (c) $3V$ free bond model.

Figure 4

Analysis of loops and bridges in $3V$ clusters. (a) Snapshot of a $3V$ cluster. The arrow head indicates a bridge. (b) Histogram of the number of bridges for a subset of clusters containing 100 particles. Bridges are present in clusters of all sizes; we focus here on clusters containing 100 particles for the sake of visualization. Concentration is 0.02, interaction energy is $8.2kT$. (c) The mean number of bridges divided by the total number of particles in clusters as a function of interaction energy. The data were collected for 50-particle (round pink) and 100-particle (square black) clusters. Concentration is 0.02. (d) The number of loops as a function of the cluster size for interaction energies 2–$101kT$. Concentration is 0.01. (e) Average number of loops per particle in clusters as a function of interaction energy.

Figure 5

Bond saturation in clusters formed by $3V$ particles. (a) Particle with free bond orientations. Cluster with fully saturated bonds. (b) Particles with fixed bond positions.

Figure 6

Minimal structural element of $3V$ clusters.

Figure 7

Decomposition of a $3V$ cluster into minimal structural elements.

Figure 8

Dependence of bond probability on interaction energy. Comparison of the bond probabilities for $3V$ models with free (black square) and fixed (red circle) bonds. Concentration is 0.01.

Figure 9

Cluster size distribution over time for $6V$ [(a),(b)] and $3V$ (c). $N=5000$. Concentration: 0.02. Interaction energy is $4kT$. Time 0 corresponds to a random distribution of monomers. (a) The cluster size distributions (symbols) of clusters larger than five particles were approximated by a Gaussian function (lines). (b) The mean ($X0$) and standard deviation $\sigma$ of Gaussian fit of (a). (c) Cluster size distributions (symbols) were approximated by a double exponential function (lines). Legends on (a) and (b) indicate the time of simulation from 0.1 to 8 s.

Figure 10

Clustering kinetics. Mean cluster size vs time for $6V$ (black line), $3V$ free bond orientation [red (gray) line], and $3V$ fixed bonds [pink (light gray) line] systems. $N=10000$ particles. Concentration is 0.01. Interaction energy is $6kT$.

Figure 11

The mean cluster size is invariant with respect to system scale. (a) Time evolution of the mean cluster size for different system scales. Number of particles ranges from 500 to 5000. Color code: pink (very light gray): 5000; magenta (light gray) = 2000; blue (gray) = 1000; black: 500 particles). Surface concentration is 0.06 and interaction energy is $6.8kT$ for all simulations. (b) Mean cluster size as a function of the system size obtained from (a). (c) Relaxation time for different concentrations. (d) Relaxation time for different interaction energies.

Figure 12

Snapshots from simulations of a system of $3V$ particles at different time points. At the start of the simulation, proteins were either randomly distributed (b) or positioned within a single centrally located cluster (a). The color code is the same as in Fig. 2. (c) Kinetics of mean cluster size for the initial conditions are shown in (a) and (b), averaged over 20 simulations. Upper blue line corresponds to initial conditions in which all particles are in one cluster; bottom red line corresponds to initial conditions in which particles are randomly distributed. Note that at steady state mean cluster size reaches the same value independent of initial conditions.