Fig. 1a: Cauliflower, a commonly available vegetable at home
Fig. 1b: Romanesco broccoli - a natural macro-size fractal (https://en.wikipedia.org/wiki/
Romanesco_broccoli)
Fractal dimension of macro-scale objects (crumpled paper balls)To demonstrate the concept at the macro scale, a simple experiment is carried out, similar to that by Gomes [3] and Amaku et al. [4], to find out the fractal dimension of crumpled paper. An A4 size 80 GSM paper is cut into pieces as shown in Fig. 2.
The individual paper pieces (see Fig. 2) are crumpled to a spherical shape as shown in Fig. 3. The mass (in g) and diameter (in mm) of these crumpled balls are measured using physical balance and vernier callipers respectively, and tabulated in Table 1. Fitting of the data from the Table 1 (see Fig.4) after taking natural logarithm of both the quantities (i.e., ln d vs. ln M) yields the following linear equation:
ln d = 3.09807 + 0.3807 * ln M
By rearranging Eq.(1), the mass and diameter of these objects are related as,
d = k M(1/df)
where, k is a constant, and df is the fractal dimension of the crumpled paper balls estimated as,
df = 1/0.3807 = 2.63
The fractal dimension obtained from this experiment is very close to the values (similar type of experiments carried out to estimate the fractal dimension of crumpled balls) reported in the literature [3]. This experiment may be repeated by different individuals to obtain statistically significant results. One can carry out this experiment with different kinds of materials such as bread, cheese, etc., as demonstrated by Amaku et al.[4] to estimate the fractal dimension.
Fractal dimension of micro- and nano-scale objects (aerosols)
On the other hand, determination of the fractal dimension in the case of micro- and nano-scale objects, such as aerosols, is not as simple. Aerosols are one among the several examples given for fractals. They are solid or liquid particles having sizes ranging from a few nm to several µm suspended in air medium. Coagulation is a phenomenon by which aerosol particles agglomerate and form clusters. Because of incomplete packing of aerosols undergoing several coagulation processes, the actual volume of a cluster is not the same as the sum of the individual solid particles forming the cluster. Practically speaking, aerosol particles may have arbitrarily irregular shape, and their non-sphericity can be described in terms of fractal geometry. For this reason, fractal dimension comes into play which is a characteristic of the type of aerosol.
One method of estimating the fractal dimension for aerosols is to compare their mass- and number-size distributions measured simultaneously. A general theory of this method which allows the particles to be fractal as well is described in the Appendix of Sapra et al.[5].
In this study, the fractal dimension is estimated from the exponent values obtained from fitting of the number- and mass- size distribution with respect to actual and aerodynamic diameters1 respectively. The mass-size distribution with respect to aerodynamic diameter was obtained by using Quartz Crystal Microscope (QCM)
while number-size distribution with respect to actual (optical) diameter is obtained using Optical Particle Counter (OPC). The fractal dimensions of incense stick smoke, room, and metallic aerosols were estimated to be 2.6, 2.56, and 1.72 respectively.
The fractal dimension of incense stick smoke and room aerosol show that their structures are close to spherical particles. The value for metallic aerosols is quite close to the known df of about 1.8 for Cluster-Cluster Aggregates (CCAs), which are known to be highly porous. The metal aerosols were produced by evaporation of metal powder in plasma torch followed by natural cooling of the vapour in the experimental chamber/vessel. The vapour nucleates to form particles that grow in the vessel due to coagulation. It follows from diffusion theory that the probability of sticking is highest at the end points of the aggregates. As a result, sharper tips continue to grow during coagulation giving rise to highly ramified branched structures, known as Diffusion Limited Aggregates (DLA). They have considerable void inside and hence a low density. DLAs, in turn, agglomerate with each other giving rise to even more branched fractal structures having lower densities. These are called Cluster-Cluster Aggregates (CCAs). The present estimate of df for metallic aerosols show that they form CCAs in dry experiments as expected, however, one needs to verify the fractal structure by direct microscopic observations.
Fig. 2: Pieces of paper used in the experiment (7 nos.)
| S.No. |
Orientation 1 |
Orientation 2 |
Orientation 3 |
Average |
Mass (M), g |
| 1 |
28.75 |
28.40 |
32.02 |
29.72 |
2.5999 |
| 2 |
23.71 |
24.68 |
26.95 |
25.11 |
1.3028 |
| 3 |
19.71 |
18.18 |
21.50 |
19.80 |
0.6401 |
| 4 |
13.14 |
15.12 |
14.84 |
14.37 |
0.3204 |
| 5 |
11.78 |
11.38 |
12.34 |
11.83 |
0.1651 |
| 6 |
7.87 |
7.91 |
8.14 |
7.97 |
0.0783 |
| 7 |
5.75 |
6.81 |
6.78 |
6.45 |
0.0410 |
Table 1: Experimental values of physical diameter and mass of crumpled paper balls shown in Fig. 3. Orientations are Diameter(d) in mm
