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

Abstract

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.

Acknowledgments

This material is based upon work supported by NASA ESPCoR under Cooperative Agreement No. NNX10AR89A, by the National Science Foundation under Cooperative Support Agreement No. EPS-0814372, and by the Desert Research Institute. Christopher Herald was partially supported by NSF Grant DMS 0709625. The authors also would like to thank Dr. Christopher Sorensen and his research group at the Kansas State University, Dr. Peter McMurry (Editor-in-Chief of Aerosol Science and Technology), and two anonymous reviewers for their helpful suggestions.