Simulation of Aggregates with Point-Contacting Monomers in the Cluster–Dilute Regime. Part 1: Determining the Most Reliable Technique for Obtaining Three-Dimensional Fractal Dimension from Two-Dimensional Images

Analysis of electron microscopy images of fractal-like aggregates involves extraction of three-dimensional (3-d) structural and geometrical properties of aggregates, which are commonly unknown, from their two-dimensional (2-d) projected images. The fractal dimension D f of an aggregate is considered to be the key property for characterizing fractal-like aggregates. The nested squares method (NSM) (also known as the cumulative-intersection method and concentric circles method), the perimeter grid method (PGM), and the ensemble method (EM) have found wide use as techniques for determination of D f of both individual and ensemble aggregates in the cluster-dilute regime. However, no study has so far compared the validity and accuracy of these three most commonly used analysis methods. In this article, using the fractal simulation package FracMAP, these methods were individually tested by applying them to a statistically significant (∼2500 per fractal dimension) number of projected images of all stable orientations of computer-generated 3-d fractal aggregates with D f ranging between 1.0 and 3.0 in increments of 0.1. Results show that of the three methods, the only method that can be used to reliably determine D f from 2-d images is the EM. Both the NSM and the PGM yield many overlapping values of 2-d D f for differing values of 3-d D f resulting in a non-one-to-one relationship and large margins of error. A correction factor has been formulated as a piece-wise function of linear functions for calibrating EM measured values of 2-d D f to actual 3-d D f values.