Advances in Condensed Matter Physics

Volume 2015, Article ID 501281, 5 pages

http://dx.doi.org/10.1155/2015/501281

## Microscale Fragmentation and Small-Angle Scattering from Mass Fractals

^{1}Joint Institute for Nuclear Research, Dubna 141980, Russia^{2}Horia Hulubei National Institute of Physics and Nuclear Engineering, 077125 Bucharest-Magurele, Romania

Received 21 April 2015; Revised 29 July 2015; Accepted 30 July 2015

Academic Editor: Fajun Zhang

Copyright © 2015 E. M. Anitas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Using the small-angle scattering method, we calculate here the mono- and polydisperse structure factor from an idealized fragmentation model based on the concept of renormalization. The system consists of a large number of fractal microobjects which are randomly oriented and whose positions are uncorrelated. It is shown that, in the fractal region, the monodisperse form factor is characterized by a generalized power-law decay (i.e., a succession of maxima and minima superimposed on a simple power-law decay) and whose scattering exponent coincides with the fractal dimension of the scatterer. The present analysis of the scattering structure factor allows us to obtain the number of fragments resulted at a given iteration. The results could be used to obtain additional structural information about systems obtained through microscale fragmentation processes.

#### 1. Introduction

The number of fragments (such as those resulted from rocks weathering and explosions or produced by earth’s crust) as a function of their sizes over a wide range of scales usually can be described by a fractal distribution [1–3], which is responsible for various physical properties, such as hydraulic conductivity or moisture characteristics in soils [4]. Although a quantification of these processes using the renormalization group approach has been suggested in [5, 6], an important issue concerns the distribution of fragments at microscales obtained by various methods such us in ultrasonic fragmentation using optoacoustic lens [7].

Small-angle scattering (SAS; neutrons, X-ray, light) [8, 9] is one of the most important techniques for investigating the microstructure of various types of systems which addresses the issue of size distribution, including the smallest and largest components. It yields the differential elastic cross section per unit solid angle as a function of the momentum transfer. The main advantage consists in its ability to differentiate between mass and surface fractals [10, 11], and it has been successfully used in studying the property of self-similarity across nano- and microscales [2], such as various types of membranes [12–15], cements [16], semiconductors [17], magnetic structures [18, 19], or biological structures [20–22]. Thus the concept of fractal geometry coupled with SAS technique can give new insights concerning the structural characteristics of such complex systems [10, 11, 23–30]. One of the main parameters which can be obtained is the fractal dimension [1]. For a mass fractal it is given by the scattering exponent of the power-law SAS intensity where . For deterministic fractals, additional information can be obtained such as scaling factor (from the period of in the logarithmic scale), the number of fractal iteration (which equals the number of periods of function ), and the total number of structural units of which the fractal is composed.

In this paper, we develop a theoretical model based on renormalization group approach which could describe various microscale fragmentation processes. We calculate analytically the mono- and polydisperse structure factor and show how to obtain the main structural parameter such as the fractal dimension and the number of fragments at a given iteration and show how to estimate the smallest and largest radius of the fractal from SAS data.

#### 2. Fragmentation Model: Construction and Properties

The construction process of three-dimensional mass fractals is similar to that of mass generalized Cantor and Vicsek fractals (GCF and GVF) [26] in the sense that one follows a top-down approach in which an initial structure is repeatedly divided (by a single scaling factor) into a set of smaller structures of the same type according to a given rule which is changed randomly from one iteration to the next one. Thus, we first consider a cube of edge length (; initiator) and in the first iteration () the number of cubes which are kept is related to the probability in which the initiator will be fragmented into 27 cubes of edge length . Figure 1 illustrates the construction of a generic model of fragmentation, where the sizes of remaining cubes at th iteration are given byAt th iteration the total number of particles will be given byand in a good approximation ( and ) it can be shown that the fractal dimension can be written as [5]with , and thus . Figure 2 shows the variation of the fractal dimension with probability in which the initiator will be fragmented into 27 cubes of edge length .