Fractal dimension of diffusion-limited aggregation clusters grown on spherical surfaces

In this work we study the fractal properties of diffusion-limited aggregation (DLA) clusters grown on spherical surfaces. Diffusion-limited aggregation clusters, or DLA trees, are highly branched fractal clusters formed by the adhesion of particles. In two-dimensional media, DLA clusters have a fractal dimension $D_f=1.70$ in the continuous limit. In some physical systems, the existence of characteristic lengths leads us to model them as discrete systems. Such characteristic lengths may result also from limitations in measuring instruments, for example, the resolution of biomedical imaging systems. We simulate clusters for different particle sizes and examine the influence of discretization by exploring the systems in terms of the relationship between the particle size $r$ and the radius of the sphere $R$. We also study the effect of stereographic projection on the fractal properties of DLA clusters. Both discretization and projection alter the fractal dimension of DLA clusters grown on curved surfaces and must be considered in the interpretation of photographic biomedical images.

Figure 1

Uniform random distribution of points on a sphere. (a) Without the transformation given by Eqs. (1) and (2), the points accumulate near the poles. (b) Using Eqs. (1) and (2), a different distribution is achieved.

Figure 2

Fractal generated in this work.

Figure 3

Stereographic projections: (a) projection A according to Eqs. (3) and (4) and (b) projection B according to Eqs. (5) and (6).

Figure 4

Plot of $M$ vs $L$ [Eq. (7)] for fractals generated with different particle sizes $r$ with (a) $r^{-1}=1024$, (b) $r^{-1}=512$, (c) $r^{-1}=256$, and (d) $r^{-1}=128$. See also Fig. 5.

Figure 5

Plot of the $D_f$ dependence on the particle size.

Figure 6

Stereographic projection A of a fractal generated on a spherical surface using different focal lengths (a) $f=-0.2$, (b) $f=-0.5$, (c) $f=-1$, and (d) $f=-2$.

Figure 7

Plot of the $D_f$ dependence on $\left|f\right|$.

Figure 8

Transition to the continuous limit according to Eq. (9) for fractals generated with different particle sizes $r$. Closed circles represent $r^{-1}=1024$ and $\beta =-0.72\pm 0.05$, open circles $r^{-1}=512$ and $\beta =-0.84\pm 0.06$, and closed squares $r^{-1}=256$ and $\beta =-1.0\pm 0.1$. Here $L$ is normalized so that $\alpha =1$ in Eq. (9).