**2.2. Models including the hydrodynamics of fluids**

In 1974 Foster and Kariapper et al. [31] performed adsorption studies on composite coatings. They proposed that if the particles in suspension acquire a positive surface charge, they could be incorporated into the metal film by electrostatic attraction. Their model is based on the following expression;

$$\frac{d\mathbf{V}p}{dt} = \frac{\mathbf{N}^\ast h \mathbf{C}\_v}{1 + h \mathbf{C}\_v} \tag{3}$$

where Vp is the volume fraction of the particles in the deposit, N\* is the number of collisions of particles suitable for the co-deposition per second and Cv is volume percent of particles in the plating bath (%).

The parameter h was related to several parameters given by the following expression;

$$h = h^\* \left( q \Delta E + L \, i^2 - ab \right) \tag{4}$$

where h\* is a constant, q is the charge density on a particle, Δ*E* is the potential field at the cathode, I is the current density, L is the bond strength of the metal/particle per surface area, a is related to the shape and b depends on agitation.

In 1987, Valdes et al [32] reported a model for co-deposition on a rotating ring disk electrode (RRDE). The equation of continuity based on differential mass balance was chosen as a starting point where different mass transport processes (Brownian diffusion and convection) for particles have been considered. According to this model, the electrochemical rate of particle deposition can be written as;

$$r\_p = K^0(\mathbb{C}\_S^\mathbb{P}) n \left[ \exp\left(\frac{aZF}{RT}\right) \eta\_a - \exp\left(\frac{1 - aZF}{RT}\right) \eta\_a \right] \tag{5}$$

Where K0 is an electrochemical rate constant which depends on *CS <sup>P</sup>*, the concentration of electroactive species (metal cations) adsorbed on the surface of the particels, n is valance and *ηa* is the activation overpotential.

In 1992, Fransaer et al. [33] proposed a trajectory model suggested for particle co-deposition on a rotating disk electrode (RDE). This particle is based on particles larger than 1 micrometer in size. In this model, Navier-Stocks equation for RDE was resolved by using Taylor expansion. The velocity of the particle that can reach towards the cathode is;

$$\frac{dz\_p}{dt} = a\_p^2 v\_{\text{stage}} \frac{F\_{\text{stage}}^{\cdot}}{F^{\cdot}} - \frac{F\_{\text{ext}}}{6\pi a\_p F^{\cdot}} \tag{6}$$

Where *Fext* is the external force at any instant along the particle, *Fstagn* ' is the force propelling the particle towards the electrode, *F* ' is the resistance force felt by the particle while it is moving perpendicular towards electrode, ap is the particle radius.

In 2000, according to the model of Vereecken et al. [34], the particles kinetics and residence time at the electrode surface have been considered. Convective-diffusion controls the transport of particles to the surface. The influence of particle gravitational force and hydrodynamics is accounted for various current densities. It is valid only when the particle size is smaller than the diffusion layer thickness. In their model they used Fick's first law and diffusion layer thickness. By combining both they got;

$$J\_p = -1.554 \,\text{w}^{-\frac{1}{6}} D\_p^{213} \left( \mathbf{C}\_{p,b} - \mathbf{C}\_{p,s} \right) \text{w}^{\frac{1}{2}} \tag{7}$$

where *CP*, *<sup>s</sup>*, is the particle concentration at the surface, *CP*,*<sup>b</sup>* is particle concentration in the bulk, Dp is the diffusion coefficient, ν is the kinematic viscosity of the solution and ω is the rotation speed of the electrode.

The ratio of the number of moles of particles to the number of moles of metal atoms equals the ratio of their fluxes Jp / Jm.

Jp can be written as;

One of the limitations of this model is that the data needed for probability coefficient are not

In 1995, a theoretical model was proposed by Fransaer et al. [30] to describe the variations in the flow of current to disk electrodes caused by a finite number of spherical and probate particles. They developed a boundary collocation method to calculate the resistance variations and the influence on current in the presence of particles and in their absence. The increase in resistance to the flow of particles was measured as a function of the diameter of the particle. The resistance is increased by larger particles to a greater degree than smaller particles. The primary current distribution around a particle was plotted as a function of the aspect ratio of the particle. The position of the particle was also determined, from its influence on the current

In 1974 Foster and Kariapper et al. [31] performed adsorption studies on composite coatings. They proposed that if the particles in suspension acquire a positive surface charge, they could be incorporated into the metal film by electrostatic attraction. Their model is based on the

> \* 1

where Vp is the volume fraction of the particles in the deposit, N\* is the number of collisions of particles suitable for the co-deposition per second and Cv is volume percent of particles in

where h\* is a constant, q is the charge density on a particle, Δ*E* is the potential field at the cathode, I is the current density, L is the bond strength of the metal/particle per surface area,

In 1987, Valdes et al [32] reported a model for co-deposition on a rotating ring disk electrode (RRDE). The equation of continuity based on differential mass balance was chosen as a starting point where different mass transport processes (Brownian diffusion and convection) for particles have been considered. According to this model, the electrochemical rate of particle

*RT RT*

h

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<sup>0</sup> <sup>1</sup> ( ) exp exp *<sup>P</sup> p S a a ZF ZF r KC n*

a

*dVp N hC*

The parameter h was related to several parameters given by the following expression;

*v v*

*dt hC* <sup>=</sup> <sup>+</sup> (3)

( ) \* 2 *h h q E Li ab* = D+ - (4)

available [29].

as it flows past the electrode.

4 Electrodeposition of Composite Materials

following expression;

the plating bath (%).

deposition can be written as;

**2.2. Models including the hydrodynamics of fluids**

a is related to the shape and b depends on agitation.

$$J\_p = \frac{\Im V\_{m,M}}{4\pi r^3 z F N\_A} \frac{\mathbf{x}\_y}{1 - \mathbf{x}\_y} \dot{\mathbf{z}}\tag{8}$$

Where *Vm*,*<sup>M</sup>* is the molar volume of metal film, *NA* is the Avagadros number, *xy* is the volume fraction of the particles in the film, r is the radius of the particles, F is Faradays constant, z is the charge of metal ions, i is the current density.

#### 6 Electrodeposition of Composite Materials

$$Jm = \frac{\dot{l}}{zF} \tag{9}$$

Combining equations 7, 8, 9;

$$\frac{\mathbf{x}\_y}{\mathbf{1} - \mathbf{x}\_y} = \frac{4\pi r^3 z FN\_A}{3V\_{m,M}} \Biggl( \mathbf{1}.554 \,\mathrm{v}^{-\frac{1}{6}} D\_p^{2l3} \Biggr) \left( \mathbf{C}\_{p,b} - \mathbf{C}\_{p,s} \right) \frac{\mathbf{w}^{\frac{1}{2}}}{\mathrm{i}} \tag{10}$$

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

As indicated in Figure 1, the model of nanoparticles co-deposition suggested by Timoshkov et al. [35], is based on the following stages:


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