Interplay of model ingredients affecting aggregate shape plasticity in diffusion-limited aggregation

We analyze the combined effect of three ingredients of an aggregation model—surface tension, particle flow and particle source—representing typical characteristics of many aggregation growth processes in nature. Through extensive numerical experiments and for different underlying lattice structures we demonstrate that the location of incoming particles and their preferential direction of flow can significantly affect the resulting general shape of the aggregate, while the surface tension controls the surface roughness. Combining all three ingredients increases the aggregate shape plasticity, yielding a wider spectrum of shapes as compared to earlier works that analyzed these ingredients separately. Our results indicate that the considered combination of effects is fundamental for modeling the polymorphic growth of a wide variety of structures in confined geometries and/or in the presence of external fields, such as rocks, crystals, corals, and biominerals.

Figure 1

Morphological diagram of clusters with 50 000 particles aggregated with different surface tension (${\alpha }_p$ values).

Figure 2

Exemplified clusters with 50 000 particles aggregated under different surface tension (${\alpha }_p$ values). Here different colors reveal growth stages, each corresponding to 1000 aggregated particles.

Figure 3

Comparison of clusters with 50 000 particles aggregated under different surface tension (${\alpha }_p$ values) for different rules for the available directions in the random walk of particles and different allowed directions for growth, for either (a) open or (b) closed boundary conditions. Here different colors reveal growth stages, each corresponding to 1000 aggregated particles.

Figure 4

Morphological diagram of clusters with 50 000 particles aggregated for different surface tension (${\alpha }_p$, on the $x$ axis) and flow (${\rho }_1$, as depicted in the inset, varied on the $y$ axis).

Figure 5

Morphological diagram of clusters with 50 000 particles aggregated for varied flow probabilities ${\rho }_1$ (left to right) and ${\rho }_2$ (top to bottom) and probabilities for the two remaining directions $\rho =(1-{\rho }_1-{\rho }_2)/2$. In each image, a sequence of clusters is shown for varied sticking probability ${\alpha }_p=0.01–0.5$, as shown in the dedicated inset.

Figure 6

Diagram of clusters with 50 000 particles aggregated for varied preferential direction of flow (angle $\mu$) and dispersion of the von Mises distribution for motion probabilities $1/\kappa$, sticking probability ${\alpha }_p=0.02$, and off-lattice model with (a) 4-neighbors or (b) 8-neighbors.

Figure 7

Schematic representation of the location parameter $\mu$ for the source of incoming particles and the importance of its choice compared to the underlying (in this case square) grid in the simulation. Here $\kappa$ is the concentration parameter of the von Mises distribution used to choose the site on the perimeter from which the particle is released.

Figure 8

Morphological diagram of clusters with 50 000 particles aggregated under different surface tension ${\alpha }_p$ and for different concentration of released particles (parameter $\kappa$), considering two locations of the particle source (i.e., angle $\mu$, marking the maximum of the distribution of the released particles on a distant circle).

Figure 9

Summarizing figure for Sec. 5, showing the influence of parameters of the nonuniform distribution of released particles and the surface tension on the obtained aggregates with 50 000 particles for the off-lattice DLA model used with either (a) 4-neighbors or (b) 8-neighbors.

Figure 10

Otoliths of zebrafish after gradual reduction of activity of the starmaker gene, from Söllner et al. [4]: (a) wild-type otolith, (b) slightly irregular otolith obtained after intermediate reduction, and (c) star-shaped otolith obtained after strong reduction.

Figure 11

Electron microscopy image of a crystal formation on an implant surface treated with calcium phosphate and acid.

Figure 12

Growth forms of the stony coral Pocillopora damicornis (from Kaandorp et al. [7]; original from Veron and Pichon [24]). The forms originate from (a) an exposed site, (b) a semiprotected site, and (c) a site sheltered from water movement.

Figure 13

Example of three series generated by the traveling observer contour decomposition (corresponding to shapes 12–14 in Fig. 14) after interpolation, reflection, and rotation. Pairwise distances among series are $d_{12–13}=0.130$, $d_{13–14}=0.070$, and $d_{12–14}=0.108$.

Figure 14

Similarity analysis of simulated aggregates obtained in this paper and experimental images of real otoliths. The tree classification (on the left-hand side) was carried out based on Ward's method [26]. Images grouped together by cluster analysis (thus shown to be similar) are displayed on the right-hand side, where pairwise distances among structures are also indicated. The parameter sets used to generate the artificial aggregates are shown in curly brackets below each image.