**3. Particle interactions and suspension stability**

#### **3.1. The DLVO theory**

 *<sup>i</sup> Jm zF*

3 1 2 6 2 3

As indicated in Figure 1, the model of nanoparticles co-deposition suggested by Timoshkov

**•** Weak adsorption of ultra-fine particles and aggregate fragments onto the cathode surface,

*<sup>x</sup> r zFN <sup>w</sup> vD C C x V i*

**•** Transport of the aggregates to the cathode surface by convection, migration and

 æ ö - <sup>=</sup> ç ÷ - - è ø

, 4 1.554 1 3 *y A l*

Where νp is the volume fraction of particles, ω is the rotation rate.

*y mM*

**•** Coagulation of ultra-fine particles in electrolytic bath,

**•** Disintegration of the aggregates in the near-cathode surface,

**•** Strong adsorption of dispersion fraction and embedment.

**Figure 1.** Model of nanoparticles co-deposition process [6, 35]

et al. [35], is based on the following stages:

**•** Formation of quasi-stable aggregates,

diffusion,

p

Combining equations 7, 8, 9;

6 Electrodeposition of Composite Materials

= (9)

1

(10)

( )

*p Pb Ps*

, ,

The state of dispersion of particles in suspension can be controlled by careful manipulation of the inter-particle forces and their interactions. A quantitative description of the relationship between stability of suspension and energies of interactions between colloidal particles and other surfaces in a liquid has been given by the classical DLVO (Derjaguin-Landau-Verwey-Overbeek) theory. According to this theory, the stability of a colloidal system is determined by the total pair interaction between colloidal particles, which consists of coulombic doublelayer repulsion and van der Waals' attraction. The total energy VT of interaction of two

$$V\_T = V\_A + V\_R \tag{11}$$

The attractive energy VA of the London-van der Waals' interaction between two spherical particles can be expressed by:

$$V\_A = -\frac{A}{6} \left( \frac{2}{S^2 - 4} + \frac{2}{S^2} \right) \tag{12}$$

where A is the Hamaker constant and S = 2 + H/a, with H the shortest distance between the two spheres and a the particle radius. If H << a, Equation (2) can be simplified to:

$$V\_A = -A \frac{a}{12H} \tag{13}$$

The repulsive energy V R is:

$$V\_R = 2\,\pi\varepsilon\varepsilon\_0\,a\,\Psi'\lim\left[1 + e^{-kH}\right] \tag{14}$$

where *ε* is the dielectric constant of the solvent, *ε* <sup>0</sup> is the vacuum dielectric permittivity, Ψ is the surface potential, l/k is the Debye length:

$$k = \left(\frac{e^2 \sum n\_i z\_i}{\varepsilon \varepsilon\_0 \, kT}\right)^{\frac{1}{2}}\tag{15}$$

where e is the electron charge, k is the Boltzmann constant, T is the absolute temperature, ni is the concentration of ions with valence zi [6].
