**2.1 Molecular dynamics approach**

The molecular dynamics description of particle-particle interaction forces shown in **Figure 3** is based on the molecular interactions through interfacial forces on the surface of the interacting particles [17]. Interfacial forces are generally assumed to act on a length scales smaller than the particle size and interactions are only possible when particles are in close proximity or during collisions. The interactions of particles in suspension depend on these surface forces which consists of the long-range ionic electrostatic repulsive forces and the short-range London-van der Waals attractive forces. Classical DLVO or colloid stability theory provides a quantitative description of the forces experienced by particles in close proximity by considering such interactions forces to be additive [18].

The magnitude of the London-van der Waals attractive force between two charged particles and the electrostatic repulsive force due to the electric double layer can be derived from their corresponding interaction potential energies [17]. Eqs. (4)

**7**

**Figure 3.**

*2019 Elsevier).*

*The Role of Micro Vortex in the Environmental and Biological Processes*

and (5) represent the magnitude of these forces while Eqs. (6) and (7) represent their respective potential energies. In addition to these surface forces, a number of other interfacial interactions such as the hydration effects, hydrophobic attraction, steric repulsion, and polymer bridging have been observed to mediate particle-particle interactions [19]. Additional forces due to the fluid-particle interactions must also be considered to fully resolve all the forces experienced by particles in suspension and

> vdw <sup>=</sup> <sup>d</sup> <sup>F</sup> dR *vdw ij*

elec <sup>=</sup> dU <sup>F</sup> dR *elec ij*

A 2R 2R 4R U 1 6 R 4R R R = ++− <sup>−</sup>

22 2 Hi i i vdw 2 22 2 ij i ij ij

<sup>2</sup> U 2 R ln 1 exp K R 2R elec i 0 ij i (7)

( ( )) = πε Ψ + − −

where *R*ij is the distance between two interacting particles (center-to-center), *A*H is the Hamaker constant, ε is the permittivity of the medium, ψ0 is the surface

In contrast to the molecular dynamics approach, the micromechanical description of particle-particle interaction relies on the geometric analysis of finite number

*Classification of different phenomena fluid-particle interactions (reproduced from with [17] permissions ©* 

potential of the particles, *K* is the reciprocal of the Debye length.

*U* (4)

*ln* (6)

(5)

*DOI: http://dx.doi.org/10.5772/intechopen.93531*

this is briefly discussed in Section 3.

**2.2 Micromechanical approach**

*The Role of Micro Vortex in the Environmental and Biological Processes DOI: http://dx.doi.org/10.5772/intechopen.93531*

*Vortex Dynamics Theories and Applications*

10–10−1 Coarse

10−2–10−4 Fine particle

10−5–10−6 Colloidal

dispersion

dispersion

dispersion

<10−6 Solution Inorganic simple

**Particle sizes, mm Classification Examples Total surface** 

Sand, mineral substances, precipitated and flocculated substances, silt, microplankton

Mineral substances, precipitated and flocculated substances, silt, bacteria, plankton, and other micro organisms

Mineral substances, hydrolyzed and precipitated products, macromolecules, biopolymers, viruses

and complex ions, molecules and polymeric species, polyelectrolytes, organic molecules and undissociated

**area, m2**

 **cm−3**

6 × 10−4–6 × 10−2 0.1–13 s

0.6–60 11 min–2 years

6 × 103 20 years

— —

**Time required to settle 100 mm if SG = 2.65**

on the chosen approach: contact forces due to the particle-particle collisions and non-contact forces due to molecular interactions at contact or interface. A brief

solutes

The molecular dynamics description of particle-particle interaction forces shown in **Figure 3** is based on the molecular interactions through interfacial forces on the surface of the interacting particles [17]. Interfacial forces are generally assumed to act on a length scales smaller than the particle size and interactions are only possible when particles are in close proximity or during collisions. The interactions of particles in suspension depend on these surface forces which consists of the long-range ionic electrostatic repulsive forces and the short-range London-van der Waals attractive forces. Classical DLVO or colloid stability theory provides a quantitative description of the forces experienced by particles in close proximity by

The magnitude of the London-van der Waals attractive force between two charged particles and the electrostatic repulsive force due to the electric double layer can be derived from their corresponding interaction potential energies [17]. Eqs. (4)

description of these forces is hereby presented in the following sections.

**2.1 Molecular dynamics approach**

*Classification of particles in dispersion [16].*

considering such interactions forces to be additive [18].

**6**

**Table 1.**

and (5) represent the magnitude of these forces while Eqs. (6) and (7) represent their respective potential energies. In addition to these surface forces, a number of other interfacial interactions such as the hydration effects, hydrophobic attraction, steric repulsion, and polymer bridging have been observed to mediate particle-particle interactions [19]. Additional forces due to the fluid-particle interactions must also be considered to fully resolve all the forces experienced by particles in suspension and this is briefly discussed in Section 3.

$$\mathbf{F}\_{\text{vdw}} = \frac{\mathbf{d}U\_{\text{vdw}}}{\mathbf{d}\mathbf{R}\_g} \tag{4}$$

$$\mathbf{F}\_{\text{alcc}} = \frac{\mathbf{d}\mathbf{U}\_{\text{dc}}}{\mathbf{d}\mathbf{R}\_y} \tag{5}$$

$$\mathbf{U}\_{\text{vobs}} = \frac{\mathbf{A}\_{\text{fl}}}{\mathbf{G}} \left[ \frac{\mathbf{z}\mathbf{R}\_{\text{i}}^{\text{z}}}{\mathbf{R}\_{\text{i}\text{j}}^{\text{z}} - \mathbf{q}\mathbf{R}\_{\text{i}}^{\text{z}}} + \frac{\mathbf{z}\mathbf{R}\_{\text{i}}^{\text{z}}}{\mathbf{R}\_{\text{i}\text{j}}^{\text{z}}} + \ln\left\{ \mathbf{z} - \frac{\mathbf{q}\mathbf{R}\_{\text{i}}^{\text{z}}}{\mathbf{R}\_{\text{i}\text{j}}^{\text{z}}} \right\} \right] \tag{6}$$

$$\mathbf{U}\_{\rm elec} = \mathbf{2}\pi\varepsilon\mathbf{R}\_{\rm i}\Psi\_{\rm o}^{\prime\prime}\ln\left[\mathbf{1} + \exp\left(-\mathbf{K}\left(\mathbf{R}\_{\rm ij} - \mathbf{2}\mathbf{R}\_{\rm i}\right)\right)\right] \tag{7}$$

where *R*ij is the distance between two interacting particles (center-to-center), *A*H is the Hamaker constant, ε is the permittivity of the medium, ψ0 is the surface potential of the particles, *K* is the reciprocal of the Debye length.

#### **2.2 Micromechanical approach**

In contrast to the molecular dynamics approach, the micromechanical description of particle-particle interaction relies on the geometric analysis of finite number

#### **Figure 3.**

*Classification of different phenomena fluid-particle interactions (reproduced from with [17] permissions © 2019 Elsevier).*

of discrete sub-elements as shown in **Figure 3**. All particle-particle interactions within this context are described by contact forces in the normal and tangential directions, while considering the elastic force-displacement, inelastic deformations or plastic dislocations, solid friction, and viscous damping [17]. Depending on the simplicity of these interactions, a soft or hard sphere description can be given. In the hard sphere model, only elastic force-displacement is allowed. Soft sphere model on the other hand allows for most of the interactions that are possible when two particles are in direct contact.

When all these contact forces are fully resolved, the behavior of the particles upon collisions or impact on a wall such as their translational and rotational velocities can be predicted with a high degree of accuracy. A detailed description of the micromechanical theory of particle collisions and its importance in the determination of particle trajectory in dispersed suspension is beyond the scope of this communication and is available elsewhere [20]. In addition to the contact forces, body forces such as gravity and buoyancy and surface forces due to the fluid are some of the other important forces acting on the particles and their quantification is highly indispensable in resolving the dynamics of particles in suspension [21]. Some of these additional forces are discussed in the next section.
