Sphractal: Estimating the Fractal Dimension of Surfaces Computed from Precise Atomic Coordinates via Box-Counting Algorithm

TL;DR

This work proposes methods to estimate the fractal dimension of the surface of any 3D object composed of spheres, by representing the surface as either a voxelized point cloud or a mathematically exact surface, and computing its box‐counting dimension.

Abstract

The fractal dimension of a surface allows its degree of roughness to be characterized quantitatively. However, limited effort is attempted to calculate the fractal dimension of surfaces computed from precisely known atomic coordinates from computational biomolecular and nanomaterial studies. This work proposes methods to estimate the fractal dimension of the surface of any 3D object composed of spheres, by representing the surface as either a voxelized point cloud or a mathematically exact surface, and computing its box-counting dimension. Sphractal is published as a Python package that provides these functionalities, and its utility is demonstrated on a set of simulated palladium nanoparticle data.

Figures (14)

• Figure 1: Illustrations of the complex atomistic surfaces of a (a) paracetamol, (b) gold-palladium nanoparticle, (c) graphene nanosheet, and (d) deoxyribonucleic acid. The white, gray, blue, red, orange, yellow, and light blue spheres correspond to hydrogen, carbon, nitrogen, oxygen, phosphorus, gold, and palladium atoms, respectively.
• Figure 2: An illustration of the dimension concept based on a scaling relationship. N represents the number of measurement units, the bold number denotes the inverse of scaling factor, while the coloured exponent indicates the dimension of the object. The first 3 iterations of the Sierpiński triangle are shown on the right-most column.
• Figure 3: Slice through illustrations of the (a) point cloud approximation and (b) voxelised representation of the surface of an octahedral nanoparticle. In the left figure, the yellow and red points lie on the outer surface (which are of interest) and inner surface (which are removed in this case, but could be retained if desired), respectively; the blue points are the centers of the surface atoms around which the yellow and red points were generated. In the right figure, the yellow boxes correspond to the voxels occupied by the yellow points in the left figure.
• Figure 4: Example output coordinates of the boxes examined over the mathematically exact surface representation of an octahedral palladium nanoparticle with increasingly smaller boxes. Each light blue sphere represents a palladium atom, while the yellow and red boxes correspond to the boxes that cover and do not cover the outer surface, respectively.
• Figure 5: Overview of Sphractal workflow. The steps coloured in red signify a common pipeline, while those coloured in green and blue correspond to the steps required only for the voxelised point cloud representation, and mathematically exact representation, respectively.