**2. Aggregates with complex structure**

An aggregate has a complex structure if it consists of smaller aggregates instead of solid monomers (Fig. 2) and their fractal dimension is different from that of the whole aggregate.

Hydrodynamic Properties of Aggregates with Complex Structure 255

Combining the last three equations, one gets the expression for the change of hydrodynamic

*B r i r*

1.56 1.728 0.228 2

*D R <sup>I</sup> <sup>r</sup>* 

which makes it possible to determine the blob hydrodynamic radius by plotting in a log-log system the number of blobs against the aggregate radius for several similar aggregates with mixed statistics and then deducing the slope and location of the best fit straight line obtained.

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.

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

The fractal dimension determined by this method for aggregates of mixed statistics

additional line are drawn in Fig. 4, representing Eq. (13) for the same fractal dimension and

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.

in the image (blob radius), which suggests very compact structure of blobs formed by

Free settling velocity of an aggregate with mixed statistics can be determined by equating the gravitational force allowing for the buoyancy of the surrounding fluid with the opposing

investigated was *D*=1.5, whereas the hydrodynamic radius of blobs 3 *Br m*

two values of the blob hydrodynamic radius, namely 2 *Br m*

Moreover, the size estimated (blob hydrodynamic radius 3 *Br m*

0

The corresponding mass-radius relation for an aggregate with mixed statistics reads

1/ 1/

*D D <sup>B</sup>*

2

(12)

(13)

. Two

. Their

and 4 *Br m*

) is close to that observed

*<sup>D</sup> <sup>D</sup>*

*B*

radius caused by the presence of blobs

**3. Asphaltene aggregates** 

system. This is presented in Fig. 4.

asphaltenes.

**4. Free settling velocity** 

Fig. 3. Typical microscope image of asphaltene aggregate.

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 hydrodynamic size down to the hydrodynamic size of blobs.

Fig. 2. Graphical representation of an aggregate with mixed statistics. The aggregate fractal dimension is a result of the spatial arrangement of blobs.

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

$$\dot{\mathbf{u}}\_B = \left(\frac{r\_B}{a}\right)^{D\_B} \tag{9}$$

$$I = \left(\frac{r}{r\_B}\right)^D \tag{10}$$

Let us imagine an aggregate of the same mass and fractal dimension composed of monomers instead of blobs. Then the total number of monomers can be expressed as

$$I\dot{\mathbf{n}}\_B = \left(\frac{r\_0}{a}\right)^D \tag{11}$$

Combining the last three equations, one gets the expression for the change of hydrodynamic radius caused by the presence of blobs

$$\frac{r}{r\_0} = i\_B \, ^{1/D\_B - 1/D} \tag{12}$$

The corresponding mass-radius relation for an aggregate with mixed statistics reads

$$I = \left[ \sqrt{1.56 - \left( 1.728 - \frac{D}{2} \right)^2} - 0.228 \right]^D \left( \frac{R}{r\_B} \right)^D \tag{13}$$

which makes it possible to determine the blob hydrodynamic radius by plotting in a log-log system the number of blobs against the aggregate radius for several similar aggregates with mixed statistics and then deducing the slope and location of the best fit straight line obtained.
