3.2 Pattern formation and analysis
Many structures formed in different processes in nature appear as similar patterns, which is sometimes explained by their common growth dynamics. For example, the flow patterns formed in our experiments result in structures similar to other natural patterns occuring in Laplacian growth systems, e.g. DLA clusters, viscous fingers in empty Hele-Shaw cells, manganese dendrites, or lightning bolts as shown in figure 3.1. In such systems, ramified structures expand at a rate proportional to the gradient of a Laplacian (∇2φ= 0) potential field, such that v ∝ ∇φ. This type of patterns is in the DLA universality class, where a fractal dimension of D= 1.71 is expected [2,40]. However, the fractal dimensions found for viscous fingering patterns in porous Hele-Shaw cells usually have values of D ∈ [1.53−1.62] [1, 2, 5]. It has been established that flow in porous media is better described by another Laplacian model, the Dielectric Breakdown Model (DBM), where the interfacial growth rate is proportional to the potential gradient of a power ηhigher than 1, i.e. v ∝(∇φ)η, where η= 2 for viscous fingers in porous media [2, 40]. A feature of pattern growth which is typical for Laplacian growth systems is that there is an active growth zone on the tips of a frozen structure due to screening of the potential gradient by the most advanced parts of the structure.
By looking closely at the patterns in figure 3.1, we notice that they have self-similar features over a range of scales, e.g. a smaller branch resemble the whole larger pattern. Another good example of self-similarity over a limited range of scales is the fern shown in figure 3.2. A common method to characterize the self-similar feature of a pattern is to analyze its fractal dimension, which is a spatial scaling exponent. In other words, if the fractal dimension of a pattern exists, it reveals information about how the pattern fills the space it occupies. There are different ways to estimate the fractal dimension of a pattern [78, 79]. Presented here are three methods we used for studying invasion patterns contained in binary images.
For a radially growing fractal pattern, its area N (mass) scales with its size r (radius) according to a power-law relationship referred to as the mass-radius relation,
N(r)∝rDm, (3.14)
where the exponent Dm is the fractal mass-dimension of the pattern. In binary images, the mass-radius relation of a radial pattern can be obtained by evaluating the area (number of white pixels) contained within a given radius from the origin of the pattern, and plot the result as function of radius in a log-log plot. The mass dimension Dm is then found as the slope of a linear fit where log10(N(r)) ∼ Dmlog10(r).
The fractal box-counting dimensionDbis found by dividing the image into boxes, i.e. dividing the image into equal squares of sides s, and count the number N of squares that contain a part of the pattern as function of box size s. For a fractal,
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A) B)
C)
D)
Figure 3.1: Dendritic patterns in different systems: A) Radial DLA simulation B) Viscous fingers in a radial Hele-Shaw cell, C) Lightning bolt, and D) Manganese dendrites.
the relationship between the number of boxes covering the pattern and their size follow a power law
N(s)∝s−Db, (3.15)
such that the box-counting dimension Db is found from slope of the linear fit log10(N(s))∼ −Dblog10(s) [2, 5, 78, 79].
The equations (3.14) and (3.15) give globally defined fractal dimensions with values between 1 and 2. Local fractal dimensions can be estimated to check how well defined these global fractal dimensions are. For a radial pattern, a local fractal dimension DL(r) is estimated as function of radius by intersecting the pattern with a concentric ring at each radius and do 1-dimensional box counts along the rings;
At each radius r, the ring is divided into equal arc segments and the number N of arc segments that intersect the pattern is counted as function of arc length l as the arc length is decreased. Similarly for linear patterns, a local fractal dimension DL(x) is estimated as function of depth along the structures; At each depth x, the structure is intersected with a perpendicular line, and 1-dimensional box counts are done along that line, i.e. the line is divided into segments of equal length l and the number N of line segments that intersect the pattern is counted as function of
Figure 3.2: It is easy to see the self-similarity of this fern. Look at the tiniest leaves of the fern: They are organized in a pattern that looks like a fern. These tiny ”ferns”
are organized on branches that look like small ferns. These fern looking branches are arranged on the stem of the fern in the familiar pattern.
segment length l as it is decreased. In both cases, for a fractal there are locally defined power laws (for given x orr)
N(l)∝l−DL, (3.16)
and we find the local fractal dimensions DL(x) orDL(r) at given x orrfrom slopes of the linear fit log10(N(l)) ∼ −DLlog10(l). For a perfect fractal, DL is constant over depth or radius.
The locally obtained fractal dimensions has values between 0 and 1. Therefore, in order to compare DL with Dm and Db it is necessary to use Mandelbrot’s rule of thumb for intersecting fractal sets [78, 80]. It states that the codimension of an intersected set equals the sum of the codimensions of the individual intersecting sets, here given by
E2−DL = (E2−E1) + (E2−D)
⇓
D=DL+ 1,
(3.17)
where D is the global fractal dimension of the pattern, E1 = 1 is the dimension of the ring or line intersecting it, and E2 = 2 is the dimension of the image plane containing the sets.