**3. Asphaltene aggregates**

254 Hydrodynamics – Advanced Topics

In opposite, the constancy of Hausdorff measure would take place in the range of the whole aggregate hydrodynamic size down to the solid monomer size. An aggregate with complex structure is termed as aggregate with mixed statistics, since it has different fractal dimensions on different length scales. The constituent aggregates are known as blobs. The knowledge of the hydrodynamic radius in relation to the radius of fractal aggregate of a given fractal dimension, utilized for blobs, makes it possible to replace the blobs by their hydrodynamic equivalents. In this way an aggregate with mixed statistics is reduced to fractal aggregate with the Hausdorff measure constant in the range of the whole aggregate

Fig. 2. Graphical representation of an aggregate with mixed statistics. The aggregate fractal

An aggregate with mixed statistics of hydrodynamic radius *r* and fractal dimension *D* consists of *I* blobs of hydrodynamic radius *Br* and fractal dimension *DB* , each containing

> *DB B*

> > *D*

*B <sup>r</sup> <sup>I</sup> r* 

> 0 *D*

Let us imagine an aggregate of the same mass and fractal dimension composed of

*Bi* solid monomers of radius *a*. The mass-hydrodynamic radius relations are

*B <sup>r</sup> <sup>i</sup> a* 

monomers instead of blobs. Then the total number of monomers can be expressed as

*B <sup>r</sup> Ii a*  *rB*

(9)

(11)

(10)

*a*

*DB*

hydrodynamic size down to the hydrodynamic size of blobs.

*D*

*r*

dimension is a result of the spatial arrangement of blobs.

The aggregate of mixed statistics can be obtained by shearing the crude oil (Gmachowski & Paczuski, 2011). Asphaltenes, a part of petroleum, are aromatic multicyclic molecules surrounded and linked by aliphatic chains and heteroatoms, of the molar mass in the range 500-50000 u. As a result of shearing, they aggregate to form blobs of a size of several micrometers, which join to form aggregates with mixed statistics. If the crude oil is mixed with toluene and n-heptane in different proportions, the range of aggregate size becomes wider.

Fig. 3. Typical microscope image of asphaltene aggregate.

It is possible to estimate the size and number of blobs for several images (Fig. 3) to identify the form of mass-radius relation of asphaltene aggregates by plotting the data in a log-log system. This is presented in Fig. 4.

The fractal dimension determined by this method for aggregates of mixed statistics investigated was *D*=1.5, whereas the hydrodynamic radius of blobs 3 *Br m* . Two additional line are drawn in Fig. 4, representing Eq. (13) for the same fractal dimension and two values of the blob hydrodynamic radius, namely 2 *Br m* and 4 *Br m* . Their locations do not correspond to points representing the experimental data, which confirms the rationality of the method of mass-radius relation for aggregates with mixed statistics.

Moreover, the size estimated (blob hydrodynamic radius 3 *Br m* ) is close to that observed in the image (blob radius), which suggests very compact structure of blobs formed by asphaltenes.
