Virtual Lab


Characterization and Properties of Hierarchical Randomly Oriented Materials (HROS)

By: Fritz Andres Campo Schickler


The prediction of the material properties need the scaling of these properties from micro-scales to macro-scales. This is a daunting task. Aerogels as well as several other materials as result of the aggregation process during their synthesis can be identified as Hierarchical Randomly Oriented Structures (HROS). These structures can be studied with Fractal Geometry.

This project shows how to characterize HROS materials through Scattering Techniques and predict macroscopic properties of the material, e.g. density and the Young modulus.

Fig.1: Through scattering experiments, either from real experiments on real materials or for a given distribution of masses in the computer, the scattering profile can be obtained. This experiment allows to fully characterizing through the whole range of wave numbers (the q-space) the structure. The FDF can be transformed from the q-space to the real space (r-space). This FDF fully characterized the structure. Through the use of Kernels the Macroscopic Response and Properties of the material can be recovered.

As depicted in the Fig. 1, there is a strong relation between the macroscopic properties, the characterization of the material structure, and the scattering response of the material. Through computational experiments the material can be studied as depicted in the Fig. 2. If the structure of the material is recreated, both, scattering or mechanical experiments can be recreated as presented in the Fig. 3,4,5 and 6 respectively.

Fig. 2: Computer experiments are performed to computer generated HROS to predict the mechanical properties. An application is the aerogel. The real pictures are taken from [1-2].

Fig. 3: In this example a Sierpinsky cube with fractal dimension 2.73 is composed by Randomly Oriented Cantor Sets with fractal dimension 0.63.

Table 1: Sepcifications of the Compound Structure consisting on a Sierpinsky cube whose primary cubes contain a Randomly Oriented Cantor Set.

Fig. 4: Fractal Dimension a s a function of the scale for the Compound Structure. There is a jump in the fractal dimension in the r-space when the scale reaches 192, the dimension of the Cantor Sets. The size of the total structure is 1728 since the Sierpinsky cube has 9 primary cubes.

Fig. 5: Computational experiment performed on the Compound structure. The inversion, in the q-space, the limiting values and the critical Limit value q are predicted by the formalism developed in this study.

In this study,

A formalism to characterize HROS has been developed.

The concept of Fractal Dimension is extended from a single number to a Fractal Dimension Function (FDF) of the scale (r-space) to fully characterize the HROS. An example is presented in the Figs. 3, 4, and 5, and the specifications of the Compound Structure are presented in the Table 1.

Computational Scattering Experiments have been performed on computer generated HROS and other hierarchical structures.

From the Scattering Intensity the fractal dimension in the wave number (q-space) can be obtained. With the formalism for HROS it can be transformed from the q-space to the r-space. An example is presented in the Fig. 3 which after transforming explains the results in the figure 4. Moreover, in the Scattering experiment presented in the Fig. 7a on real aerogels, allows to find the characteristic FDF in the q-space which after the transformation gives the FDF in the r-space depicted in the Fig. 7b. With this, the density is calculated as presented in the Fig. 7b.

Using a Kernel which relates the macroscopic property with the FDF, the macroscopic properties have been recovered.

The density can be predicted from the FDF finding similar values for the real material as result of the scattering experiments as presented in the Fig. 7b.

Fig. 6: Mechanical Tests are performed on computer generated HROS to characterize them.

a) b)

Fig. 7: a) For the experimental data on aerogels [3] the FDF in the q-space is obtained (D(q)) from the Scattering Profile I(q). b) FDF in the r-space (D(r)) is obtained from a). In addition, the density as a function of scale is calculated.


[1] Robert M.Brown. JLP. Los Alamos National Laboratories.

[2] A. Tscheschel, J. Lacayo and D. Stoyan .Characterization of Filled Elastomers with Methods of Spatial Statistics. Poster TU Bergakademie Freiberg, Continental AG, Hannover.

[3] T. Woigner et al. Different Kinds of Fractal Structures in Silica Aerogels. J. Non Crystalline Solids. 121, (1990)198-201.