Segregation of fractal aggregates grown from two seeds

We study the generalized diffusion-limited aggregates, grown from two proximal nucleation seeds placed at distance $d$ lattice units and investigate the probability $p\left(d\right)$ that these aggregates get connected. We vary the sticking probability to get a range of aggregate geometry from fractal to compact one. For fractal aggregates, $p\left(d\right)$ decays rapidly with $d$, while for compact ones, the decay is so slow that $p\left(d\right)\approx 1$ for all practical distances. We experimentally demonstrate similar behavior for viscous fingering patterns with two injection points and electrochemical deposits grown on two cathodes. Our observations along with previous results on competitive growth suggest a common underlying principle.

Figure 1

Representative morphologies of the GDLA patterns generated using two seed configuration for $d=9$ lattice units. (a) $\alpha =1.0$; $D_F=1.67$, (b) $\alpha =0.7$; $D_F=1.74$, and (c) $\alpha =0.3$; $D_F=1.84$.

Figure 2

Plots shows (a) probability of connectivity $P$ $(d,N)$ versus number of particles $N$ plotted for $d=2$ lattice units ($\alpha =1.0$ [엯]; 0.7 [◇]; 0.4 [▵]; 0.1 [▿]) and $d=9$ lattice units ($\alpha =0.3$ [⋯]; 0.2 [----]; 0.1 [—]), (b) asymptotic probability of connectivity $p\left(d\right)$ against interseed distance $d$ for various $\alpha$ values and (c) variation of $P(2,N)$, against $N$ for 3D clusters generated using two-seed configuration on cubic lattice.

Figure 3

Connectivity phase diagram for two-point injection viscous fingering in radial Hele-Shaw cell for air-glycerine system. The line is drawn to guide the eye for separation between two regimes.

Figure 4

Zn electrodeposits using aqueous $\mathrm{Zn}\mathrm{S}{\mathrm{O}}_4$ solution for two-cathodes placed at a distance $0.2\mathrm{mm}$ and both are connected to a same $-5.6\mathrm{V}$ with respected to ring anode. Segregated and connected patterns are observed for electrolyte concentrations (a) $0.01\mathrm{M}$, (b) $0.02\mathrm{M}$, respectively. Two wire electrodes $(\text{diameter}=0.2\mathrm{mm})$ can be clearly seen in photograph. Horizontal scale bar is $1\mathrm{mm}$ long