**1. Introduction**

250 Hydrodynamics – Advanced Topics

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Hydrodynamic properties of fractal aggregates and polymer coils, such as sedimentation velocity, permeability, translational and rotational diffusion coefficients and intrinsic viscosity, are of great interest in hydrodynamics, engineering, colloid and polymer science and biophysics. The hydrodynamic properties of aggregates are closely connected to their structure.

Aggregates – the clusters of monomers - usually have a fractal structure which means that parts of the object are similar to the whole. The self-similar structure is characterized by the fractal dimension which is a measure of how the aggregate fills the space it occupies. The fractal dimension can be calculated by analyzing the mass-radius relation for a series of similar aggregates, since the mass of an aggregate scales as a power of the size.

The fractal dimension can be also determined by covering the aggregate with spheres of changing radius (Fig. 1). Then plotting the number of spheres *N* versus their radius in a log-log coordinate system, one determines the fractal dimension as the negative slope of the obtained line.

Fig. 1. Aggregate covering with spheres of changing radius.

Hydrodynamic Properties of Aggregates with Complex Structure 253

For similar aggregates the size of large pores scales as the size of the whole aggregate. Therefore the ratio of the internal permeability and the square of aggregate radius is expected to be constant for aggregates of the same fractal dimension and to decrease with increasing fractal dimension due to the increment of the aggregate compactness (Gmachowski, 1999; Woodfield & Bickert, 2001; Bushell et al., 2002). This means that

*R k* / is a unique function of the fractal dimension of an aggregate and hence the ratio

has been derived from analysis of permeability of aggregated system (Gmachowski, 2000) and is confirmed by different hydrodynamic properties of fractal aggregates (Gmachowski,

This means that the hydrodynamic radius is proportional to aggregate radius for a given fractal dimension. The covering can be thus performed not only in the range of radii, but also in the range of hydrodynamic radii. The hydrodynamic radius of a solid monomer is its geometrical radius. For fractal aggregate the mass-hydrodynamic radius relation has the

*D*

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

since the hydrodynamic radius *r* converges to the primary particle radius *a* for the number of constituent particles equal to unity (Gmachowski, 2008). The mass-radius relation reads

> *D D r R <sup>i</sup> R a*

which reduces to the previous one if the aggregation number is related to the hydrodynamic

1.56 1.728 0.228 2

*<sup>D</sup> D R <sup>i</sup>*

Plotting in a log-log system the aggregation number against radius for several similar aggregates, one can determine a best fit straight line whose slope is the fractal dimension

If the aggregate is composed of smaller aggregates instead of solid monomers, their number is correlated to the hydrodynamic radius of smaller aggregates according to the mass-radius

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.

2

*D*

*a*

radius instead to the radius. The full form of mass-radius relation has the form

and the location makes it possible to determine the monomer radius.

relation of the form similar to Eq. (8).

**2. Aggregates with complex structure** 

*r D*

*R*

1.56 1.728 0.228 2

2

(5)

(6)

(7)

(8)

2003).

following form

*r/R* is determined by *D* (Eq. 4). A formula

The Hausdorff dimension (Hausdorff, 1919) is the critical exponent for which the Hausdorff measure *Md* , being proportional to the product of number of spheres and a power of their radius, changes from zero to infinity when the size of covering elements tends to zero

$$M\_d \propto N\left(\rho\right) \cdot \rho^d \xrightarrow{\rho \to 0} \begin{cases} 0, & d > D \\ \infty, & d < D \end{cases} \tag{1}$$

In practice each monomer has its size. It is thus generally accepted the constancy of Hausdorff measure in a finite range of size to be sufficient to characterize the aggregate structure. The constancy of the Hausdorff measure for two limiting sizes of spheres, written for an aggregate containing i monomers, can be expressed as

$$\mathbf{1} \cdot \mathbf{R}^D = \mathbf{i} \cdot \mathbf{R}\_1^D \tag{2}$$

where R is the radius of the sphere circumscribed on the aggregate and *R*<sup>1</sup> is the radius of envelope surrounding one monomer, for which the similarity to the aggregate still exists (Gmachowski, 2002).

The structure of aggregates is permeable which means that a fluid flows not only around but also through the aggregate. It is analyzed by taking into account the internal permeability of aggregates, either directly or by replacing a given aggregate by a smaller impermeable sphere of the same hydrodynamic properties. In this way the hydrodynamic radius is defined.

The structure of fractal aggregate can be related to the possibility to penetrate its interior by a fluid, well represented by internal permeability. The translational friction coefficient of a particle of radius R can be written in the following form

$$f\_T = 6\pi\eta\_0 R \cdot \frac{r}{R} \tag{3}$$

where the hydrodynamic radius *r* is introduced to take into account its dependence on the internal permeability of the aggregate. Such relation gave Brinkman for translational friction factor (Brinkman, 1947)

$$\frac{r}{r\_R} = \frac{1 - \frac{\tanh\sigma}{\sigma}}{1 + \frac{3}{2\sigma^2} \left(1 - \frac{\tanh\sigma}{\sigma}\right)}\tag{4}$$

where *R k* / is the reciprocal square root of dimensionless internal permeability of a sphere of uniform structure modeling the fractal aggregate. The analogous relations of the normalized hydrodynamic radius for the rotational friction coefficient and the intrinsic viscosity are slightly different, but all the three give the results which are very close to one another (Gmachowski, 2003).

For a homogeneous porous medium, being an arrangement of monosized particles, the permeability is proportional to the square of the characteristic pore size (Dullien, 1979) which is closely correlated to the size of constituents. In the case of fractal aggregate, which is not homogeneous, the fluid flow occurs mainly in the large pores. Hence their size determines the aggregate permeability.

The Hausdorff dimension (Hausdorff, 1919) is the critical exponent for which the Hausdorff measure *Md* , being proportional to the product of number of spheres and a power of their

radius, changes from zero to infinity when the size of covering elements tends to zero

*d*

 

*d*

*M N*

for an aggregate containing i monomers, can be expressed as

particle of radius R can be written in the following form

(Gmachowski, 2002).

radius is defined.

where

friction factor (Brinkman, 1947)

another (Gmachowski, 2003).

determines the aggregate permeability.

<sup>0</sup> 0,

In practice each monomer has its size. It is thus generally accepted the constancy of Hausdorff measure in a finite range of size to be sufficient to characterize the aggregate structure. The constancy of the Hausdorff measure for two limiting sizes of spheres, written

where R is the radius of the sphere circumscribed on the aggregate and *R*<sup>1</sup> is the radius of envelope surrounding one monomer, for which the similarity to the aggregate still exists

The structure of aggregates is permeable which means that a fluid flows not only around but also through the aggregate. It is analyzed by taking into account the internal permeability of aggregates, either directly or by replacing a given aggregate by a smaller impermeable sphere of the same hydrodynamic properties. In this way the hydrodynamic

The structure of fractal aggregate can be related to the possibility to penetrate its interior by a fluid, well represented by internal permeability. The translational friction coefficient of a

> <sup>0</sup> 6 *<sup>T</sup> <sup>r</sup> f R <sup>R</sup>*

where the hydrodynamic radius *r* is introduced to take into account its dependence on the internal permeability of the aggregate. Such relation gave Brinkman for translational

tanh <sup>1</sup>

*R k* / is the reciprocal square root of dimensionless internal permeability of a

3 tanh 1 1

sphere of uniform structure modeling the fractal aggregate. The analogous relations of the normalized hydrodynamic radius for the rotational friction coefficient and the intrinsic viscosity are slightly different, but all the three give the results which are very close to one

For a homogeneous porous medium, being an arrangement of monosized particles, the permeability is proportional to the square of the characteristic pore size (Dullien, 1979) which is closely correlated to the size of constituents. In the case of fractal aggregate, which is not homogeneous, the fluid flow occurs mainly in the large pores. Hence their size

2

2

*r R*

,

*d D*

*d D*

<sup>1</sup> 1 *D D R iR* (2)

(3)

(4)

(1)

For similar aggregates the size of large pores scales as the size of the whole aggregate. Therefore the ratio of the internal permeability and the square of aggregate radius is expected to be constant for aggregates of the same fractal dimension and to decrease with increasing fractal dimension due to the increment of the aggregate compactness (Gmachowski, 1999; Woodfield & Bickert, 2001; Bushell et al., 2002). This means that *R k* / is a unique function of the fractal dimension of an aggregate and hence the ratio *r/R* is determined by *D* (Eq. 4). A formula

$$\frac{r}{R} = \sqrt{1.56 - \left(1.728 - \frac{D}{2}\right)^2} - 0.228 \tag{5}$$

has been derived from analysis of permeability of aggregated system (Gmachowski, 2000) and is confirmed by different hydrodynamic properties of fractal aggregates (Gmachowski, 2003).

This means that the hydrodynamic radius is proportional to aggregate radius for a given fractal dimension. The covering can be thus performed not only in the range of radii, but also in the range of hydrodynamic radii. The hydrodynamic radius of a solid monomer is its geometrical radius. For fractal aggregate the mass-hydrodynamic radius relation has the following form

$$i = \left(\frac{r}{a}\right)^D\tag{6}$$

since the hydrodynamic radius *r* converges to the primary particle radius *a* for the number of constituent particles equal to unity (Gmachowski, 2008). The mass-radius relation reads

$$i = \left(\frac{r}{R}\right)^D \left(\frac{R}{a}\right)^D \tag{7}$$

which reduces to the previous one if the aggregation number is related to the hydrodynamic radius instead to the radius. The full form of mass-radius relation has the form

$$\dot{q} = \left[ \sqrt{1.56 - \left( 1.728 - \frac{D}{2} \right)^2} - 0.228 \right]^D \left( \frac{R}{a} \right)^D \tag{8}$$

Plotting in a log-log system the aggregation number against radius for several similar aggregates, one can determine a best fit straight line whose slope is the fractal dimension and the location makes it possible to determine the monomer radius.

If the aggregate is composed of smaller aggregates instead of solid monomers, their number is correlated to the hydrodynamic radius of smaller aggregates according to the mass-radius relation of the form similar to Eq. (8).
