Arc-length characterization of finite, radial growth patterns

Abstract

We present a method to characterize the distribution of length-scales of finite, disordered patterns with, on average, radial symmetry. This method makes it possible to quantify the distribution of characteristic length scales in cases where the conventional "linear" chord method does not work. We show that the method can clearly distinguish regular patterns, patterns that are formed by diffusion-limited aggregation, and patterns that form during the slow drying of confined, colloid-laden droplets, explained by Beechey-Newman et al.1 We also introduce a method to find the centre-point of these finite patterns, without assuming a full connectivity in the pattern. The method should be widely applicable to other, finite quasi-two-dimensional patterns.

Figures (8)

• Figure 1: Illustration of the experimental setup from ref. Beechey-Newman_etal_2025 that was used to obtain the colloidal maze-like patterns. A microscope slide is prepared with a double-sided sticky tape, where we punch a hole of $r=6$ mm and deposit a droplet of water containing 1 wt.% of 1 $\mu$m large TPM-particles, as shown in (a). Then, a cover-slip is placed on top of the circular cell, forcing the evaporation process to take many days. A cross-section of the cell is shown in (b), showing the colloid deposition both on the bottom and top glass plates. The slow evaporation process drives colloids to assemble into a contiguous maze, as seen in the photograph shown in (c).
• Figure 2: (a) Illustration of the circular arc-length distribution for the colloidal drying pattern P2, that was appropriately cropped and binarized. For a given radius (red circle), each void-solid (white-black) interface is marked with a red dot, allowing us to store the length along the circle between each interface. (b) Resulting distribution of the arc lengths $s$ between interfaces, where $p(s)$ is the probability for a given $s$. Radii lying in the orange shell show a broader distribution than those lying in the blue, inner shell, with finer structures. (c) Telegraph signal $\mathcal{T}(\theta)$ obtained from the particular radius illustrated in (a), with $\theta$ starting from the 3 o' clock position moving clockwise. In the signal, 1 indicate the solid phase and 0 corresponds to the void phase. The details of the CAL distribution differ from sample-to-sample, but the overall characteristics remain the same.
• Figure 3: The 5 patterns we compare in this paper, where 3 are drying patterns, obtained by the method described in Figure 1 Beechey-Newman_etal_2025. The images were then appropriately cropped and binarized. The lower two images are that of a DLA pattern, which was generated with 400000 particles in a $2048 \times 2048$ grid and a RBP pattern, which produces a radially invariant CAL distribution.
• Figure 4: Average arc-lengths of both phases, averaged over $dr=20$ pixels, for the 3 drying patterns P1-P3. The growth trends of the synthetic patterns are added as dashed lines for illustration.
• Figure 5: Solid fraction $\varphi$ along the circular shell of $dx=1$ at radius $r$. 1 means the shell contains only solid, e.g. at the centre of the patterns, while 0 meaning the shell contains only void phase.