Kinetic percolation

We demonstrate that kinetic aggregation forms superaggregates that have structures identical to static percolation aggregates, and these superaggregates appear as a separate phase in the size distribution. Diffusion limited cluster-cluster aggregation (DLCA) simulations were performed to yield fractal aggregates with a fractal dimension of 1.8 and superaggregates with a fractal dimension of $D=2.5$ composed of these DLCA supermonomers. When properly normalized to account for the DLCA fractal nature of their supermonomers, these superaggregates have the exact same monomer packing fraction, scaling law prefactor, and scaling law exponent (the fractal dimension) as percolation aggregates; these are necessary and sufficient conditions for same structure. The size distribution remains monomodal until these superaggregates form to alter the distribution. Thus the static percolation and the kinetic descriptions of gelation are now unified.

Figure 1

Aggregate monomer count $N$ vs aggregate radius of gyration $R_g$ normalized by monomer radius $a$ on log-log plots. Linear regimes imply the power law as described in Eq. (2); the slope is the exponent, the fractal dimension $D$. All systems start at small $R_g/a$ with $D=1.8$ and a prefactor $k_0=1.35$. In (a) the initial monomer fraction was $f_v=0.003$ and the system reached a $D=2.5$ with a prefactor of $k_0=0.05$ regime at large $R_g/a$. In (b) $f_v=0.01$ and reached $D=2.5$ with $k_0=0.10$. In (c) $f_v=0.02$ and reached $D=2.5$ with $k_0=0.15$. In (d) $f_v=0.1$ and reached $D=2.5$ with $k_0=0.50$. In all plots the average supermonomer size $\langle R_0\rangle$ is marked.

Figure 2

In (a) $R_g$ data from Fig. 1 are binned and the average $N$ for each bin is plotted for clarity. The data in (a) indicate that small aggregates, i.e., $N<10–30$, cannot differentiate themselves between $D=1.8$ and $D=2.5$. In (b) the $R_g$ values from (a) are normalized by the supermonomer radius $\langle R_0\rangle$, and $N$ is normalized by the supermonomer count $\langle N_0\rangle$. These renormalized plots now all fall onto a single trend. Furthermore, the trend overlaps that for aggregates made via the simple, cubic lattice, static percolation algorithm. The fit to Eq. (2) for all these yields $D=2.5$ and $k_0=0.6$.

Figure 3

The pair correlation function $g\left(r\right)$ for aggregates (circles symbols) taken from DLCA simulations with fits lines derived from Eqs. (6) and (8). The monomer packing fraction $\phi =0.7$ is found for all DLCA aggregates below the superaggregate crossover between $\langle R_0\rangle$ and $R_{g,G}$. In (a) an aggregate taken from a DLCA system with $f_v=0.001$ that has not developed into superaggregates. For (b)–(e), $g\left(r\right)$ of aggregates with $f_v=0.003,0.01,0.02$ and 0.1, respectively, the average supermonomer size $\langle R_0\rangle$ and radius of gyration at the gel point $R_{g,G}$ are marked. The crossover from $D=1.8$ to $D=2.5$ is flanked by $\langle R_0\rangle$ on the left and $R_{g,G}$ on the right at larger $r$. In (f) the $g\left(r\right)$ of a static percolation aggregate is plotted and is described by ${\phi }_0=0.3$ and $D=2.5$ which matches the DLCA superaggregates well.

Figure 4

Particle count spectra (upper plots) and particle mass spectra (lower plots) for two monomer volume fractions $f_v=0.01$ and 0.02. $N_c$ is the total number of clusters remaining in the aggregation run.