**2.1 Kinetic models**

44 Ceramic Coatings – Applications in Engineering

experimental results, but the current experts opinion is that a full understanding is still lacking, mainly due to the phenomena that are at the base of the interaction between the charged particles approaching both each other and to the electrode to form the solid deposit. For some applications, a requirement is that the ceramic EPD deposit is dense, so a postdeposition treatment should be performed in order to densify it. Usually, this consists of a conventional heating treatment in a furnace, but some problems could occur such as delamination, cracks or residual stress due to the differential coefficient of thermal expansion. Moreover, the high densification temperature of ceramics can be detrimental for

Sometimes, the sintering temperature can be decreased by adding some low melting additives. Alternative sintering methods could be considered, such as microwave, laser or

Finally, the wide range of applications of EPD deposits will be mentioned. EPD process is very versatile, therefore porous, layered, and graded deposits can be obtained besides dense coatings. Recently, it has been clearly demonstrated the possibility of obtaining nanocomposite materials, especially those containing carbon nanotubes (CNTs), by using EPD. As a consequence, the applications are in a spread number of sectors: biomaterials, fuel

Electrophoretic Deposition is a traditional processing method in the ceramic industry that is

EPD is achieved through the movement of charged particles dispersed in a suitable liquid towards an electrode under an applied electric field. This movement results in the accumulation of the particles and in the formation of a homogeneous deposit at the

The main requirement to obtain an efficient EPD process is to use suitable suspensions where ceramic particles are well suspended and dispersed. When a ceramic particle is in a liquid medium, it can be charged through four mechanisms (Van der Biest & Vandeperre, 1999):

d. electron transfer between the solid and the liquid phase due to the difference in work

For the analysis and discussion on charging mechanisms and particles interactions, that are at the base of this ceramics processing method, one can refer to the fundamentals of colloid

Since EPD is assumed to be a two-step process, electrophoresis and deposition, each step require accurate attention. Firstly, the kinetics of the process will be described giving particular attention to the relation between process parameters and time evolution of the deposit yield. Then, the mechanisms of deposition proposed in literature will be discussed.

the substrate which could be damaged.

**2. Fundamentals and models** 

appropriate electrode (Figure 1).

function.

cells, barrier coatings, electronics, catalysis, optical devices.

gaining increasing interest for production of new materials coatings.

a. selective adsorption of ions onto the solid particle from liquid,

c. adsorption or orientation of dipolar molecules at the particle surface, and

b. dissociation of ions from solid phase into the liquid,

science widely discussed in literature (Lewis, 2000).

electron beam.

The first model used to describe EPD process is attributed to Hamaker (Hamaker, 1940) who proposed a general expression for the deposited mass per area unit (*m,* g cm-2) in a cell with planar geometry:

$$
\mu m = \mathcal{C}\_s \mu AE t \tag{1}
$$

where CS, solids concentration in the suspension (g cm-3); t, deposition time (s); µ, electrophoretic mobility (cm2 V-1 s-1); E, electric field strength (V cm-1); A, surface area of the electrode (cm2).

Several years later, Sarkar and Nicholson (Sarkar & Nicholson, 1996) considered again the first model of Hamaker and analysed the dependence of kinetics on some experimental conditions.

In eq. (1), µ and A can be evaluated numerically and one reasonably supposes that they are constant during the process. It is not so true for CS and E, that vary as the process while the process is going on.

Sarkar and Nicholson considered the variation of the particles concentration in the suspension for long deposition time, starting with the condition that the only change in the concentration is due to the mass of powder deposited by EPD. It is equal to zero when the process starts and varies with time according to the expression:

$$m(t) = m\_0(1 - e^{-t/\tau})\tag{2}$$

where m0, initial mass of powder in suspension (g) and � � � ��� ⁄ defined as characteristic time (V is the volume of the suspension considered constant). Actually, is the reciprocal of k, the "kinetic parameter" that represents a key parameter in the modelling of EPD process.

Ceramic Coatings Obtained by Electrophoretic Deposition:

above mentioned.

constant current density:

of EPD process.

main current carriers.

Fundamentals, Models, Post-Deposition Processes and Applications 47

The hypotheses and the conditions that are at the basis of the eq. (3) were widely discussed by Van der Biest (Van der Biest & Vandeperre, 1999), who cited also much experimental evidence on the dependence of the electric field in the suspension from the parameters

In order to avoid the effect of deposit resistivity increasing, Sarkar and Nicholson proposed to work in galvanostatic condition rather than in potentiostatic condition. In such a way the number of particles arriving on the electrode is constant and not influenced by the electrical condition of the suspension-deposit system. Ma (Ma & Cheng, 2002) determined the relationship between the kinetic parameter of Sarkar and Nicholson and the applied current density, *i*, making the predictions of the deposition yield easier, when the deposition is at

Other factors that influence the drop of the electric field are a change in the polarization of the electrode and a change in the conductivity of the suspension, occurring in the progress

Changes in the electrode polarization can be due to a change in the concentration of the reactants at the electrode-electrolyte interface. When the process starts and a current flows towards the electrode, the concentration of reacting species at the electrode drops off. They are replaced by the species of bulk suspension through diffusion, convection or migration, but the presence of deposit layer can retard the transport of reactants to the electrode. As a

As regards the suspension conductivity, a model introduced by Vandeperre and Van der Biest (Vandeperre & Van der Biest, 1998) affirms that both the particles and the ions surrounding the particles contribute to the conductivity. During the EPD process, the electric conductivity decreases due to the effect of the depletion of particles in the suspension and for the reduced presence of ions moving together with powder particles. This effect is more evident if the amount of deposited powder is high with respect to the

Several studies were devoted to the investigation of the effect of suspension conductivity on the EPD process, through the study of the influence of binders, salts, stabilizers, and a liquid medium (aqueous and non aqueous)(Ferrari & Moreno, 1996; Westby at al., 1999; Moreno & Ferrari, 2000; De Riccardis et al., 2007). As a common result, in order to have an effective deposition process the electrostatic or electrosteric stabilisation of the suspension has to be such that the ionic concentration in suspension is low and the suspended particles are the

In order to schematize the EPD yield and according to Sarkar and Nicholson, it is possible to recognize four different behaviours, depending on the process conditions: constant voltage,

In Curve A of Fig. 3 (constant current and constant concentration) the deposition rate is linear with respect to time. In curves B, C and D, the deposition rate decreases

powder in suspension and if the suspension ionic conductivity is relatively small.

constant current, constant concentration, and variable concentration.

where *k0*, a reference kinetic constant, and *i0*, a reference applied current density.

consequence, the polarization at the working electrode changes.

������� � <sup>⁄</sup> � − 1) (4)

In order to consider that some particles arriving to the electrode do not take part to the formation of the deposit, Sarkar and Nicholson introduced a "sticking factor", f ≤ 1, a multiplicative efficiency factor.

For short time, eq. (2) is reduced to the Hamaker model, since in the early stages of the process the variation of bulk concentration is negligible due to the small amount of powder deposited. When the process advances, the deposited mass is relevant and its effect on the evolution of the deposition process is not negligible. In this condition, under the hypothesis that the resistivity of deposited layer is higher than that of the suspension, the electric field suffers a drop, also if the process is conducted under the condition of constant applied voltage.

In eq. (1), the electric field strength is the effective strength that affects the particles in the suspension and so it is decreased by the potential drops due to the resistance of electrodes, suspension and deposited layer. With respect to the applied voltage externally, *v*, the expression of the electric field experienced by the particles is:

$$E(t) = \left(\upsilon - \left[\Delta V\_{e1} + iR\_d \mathbf{s}(t) + iR\_s \{d - \mathbf{s}(t)\} + \Delta V\_{e2}\right]\right) / d\tag{3}$$

*Ve1* and *Ve2* are the potential drop (V) at the electrode 1 and 2, respectively; *i*, is the current (A); *d* is the distance between the electrodes (m); *s(t)* is the thickness of the deposited layer (m); *Rd* and *Rs* are the resistivity (/m) of the deposit and the suspension (Fig. 2).

Fig. 2. Variation of the potential in an electrophoretic cell during the deposition

In order to consider that some particles arriving to the electrode do not take part to the formation of the deposit, Sarkar and Nicholson introduced a "sticking factor", f ≤ 1, a

For short time, eq. (2) is reduced to the Hamaker model, since in the early stages of the process the variation of bulk concentration is negligible due to the small amount of powder deposited. When the process advances, the deposited mass is relevant and its effect on the evolution of the deposition process is not negligible. In this condition, under the hypothesis that the resistivity of deposited layer is higher than that of the suspension, the electric field suffers a drop, also if the process is conducted under the condition of constant applied

In eq. (1), the electric field strength is the effective strength that affects the particles in the suspension and so it is decreased by the potential drops due to the resistance of electrodes, suspension and deposited layer. With respect to the applied voltage externally, *v*, the

(A); *d* is the distance between the electrodes (m); *s(t)* is the thickness of the deposited layer

(m); *Rd* and *Rs* are the resistivity (/m) of the deposit and the suspension (Fig. 2).

Fig. 2. Variation of the potential in an electrophoretic cell during the deposition

�(�) � (� � ����� � ����(�) � ����� � �(�)� � �����)�� (3)

*Ve2* are the potential drop (V) at the electrode 1 and 2, respectively; *i*, is the current

expression of the electric field experienced by the particles is:

multiplicative efficiency factor.

voltage.

*Ve1* and

The hypotheses and the conditions that are at the basis of the eq. (3) were widely discussed by Van der Biest (Van der Biest & Vandeperre, 1999), who cited also much experimental evidence on the dependence of the electric field in the suspension from the parameters above mentioned.

In order to avoid the effect of deposit resistivity increasing, Sarkar and Nicholson proposed to work in galvanostatic condition rather than in potentiostatic condition. In such a way the number of particles arriving on the electrode is constant and not influenced by the electrical condition of the suspension-deposit system. Ma (Ma & Cheng, 2002) determined the relationship between the kinetic parameter of Sarkar and Nicholson and the applied current density, *i*, making the predictions of the deposition yield easier, when the deposition is at constant current density:

$$k = k\_0(e^{l/l\_0} - 1)\tag{4}$$

where *k0*, a reference kinetic constant, and *i0*, a reference applied current density.

Other factors that influence the drop of the electric field are a change in the polarization of the electrode and a change in the conductivity of the suspension, occurring in the progress of EPD process.

Changes in the electrode polarization can be due to a change in the concentration of the reactants at the electrode-electrolyte interface. When the process starts and a current flows towards the electrode, the concentration of reacting species at the electrode drops off. They are replaced by the species of bulk suspension through diffusion, convection or migration, but the presence of deposit layer can retard the transport of reactants to the electrode. As a consequence, the polarization at the working electrode changes.

As regards the suspension conductivity, a model introduced by Vandeperre and Van der Biest (Vandeperre & Van der Biest, 1998) affirms that both the particles and the ions surrounding the particles contribute to the conductivity. During the EPD process, the electric conductivity decreases due to the effect of the depletion of particles in the suspension and for the reduced presence of ions moving together with powder particles. This effect is more evident if the amount of deposited powder is high with respect to the powder in suspension and if the suspension ionic conductivity is relatively small.

Several studies were devoted to the investigation of the effect of suspension conductivity on the EPD process, through the study of the influence of binders, salts, stabilizers, and a liquid medium (aqueous and non aqueous)(Ferrari & Moreno, 1996; Westby at al., 1999; Moreno & Ferrari, 2000; De Riccardis et al., 2007). As a common result, in order to have an effective deposition process the electrostatic or electrosteric stabilisation of the suspension has to be such that the ionic concentration in suspension is low and the suspended particles are the main current carriers.

In order to schematize the EPD yield and according to Sarkar and Nicholson, it is possible to recognize four different behaviours, depending on the process conditions: constant voltage, constant current, constant concentration, and variable concentration.

In Curve A of Fig. 3 (constant current and constant concentration) the deposition rate is linear with respect to time. In curves B, C and D, the deposition rate decreases

Ceramic Coatings Obtained by Electrophoretic Deposition:

electrical resistance of the suspension is constant.

counter electrode, DL= double layer)

suspension is negligible.

Fundamentals, Models, Post-Deposition Processes and Applications 49

constant, c) the electrical resistance of the EPD deposit is proportional to its thickness, d) the

Fig. 4. Electric model of EPD process with plane electrode (we= working electrode, ce=

electrode/suspension, but it is less relevant as no deposition occurs at that electrode.

The assumption a) is based on the estimate of time required for double-layer charging. The assumption b) is supported by the consideration that after the formation of the first monolayer, the interface at the working electrode/suspension should be stabilised as the charging of double-layer should be completed. Similarly at the interface between counter

The assumption c) relies on the compliance of the physical properties of deposited layer comparable with a porous media, where the electrical resistance is proportional to its thickness. The assumption d) is based on the consideration that the deposited layer is less thick than the electrodes distance, therefore the decrease in the volume filled by the

Baldisserri applied this model to EPD of TiO2 particles and verified experimentally that the previously mentioned assumptions were satisfied. As a result, they derived a linear correlation between deposited mass and passed charge and a non-linear kinetic equation:

�(� � ��)��� � ��, � �

���� � � � �� ��

� (5)

�(�) � � ���

����

asymptotically with deposition time. Obviously, the decreasing concentration produces a reduction of the final yield and therefore of the rate deposition either at constant current (Curve B) or at constant voltage (curve C and D).

Comparing curve A and C (both with constant concentration), the final yield is considerably higher in curve A than in curve C. This is due to the electric resistance of the deposit layer, considered higher than that of an equal thickness taken up by the suspension. Therefore, when the process goes on, the voltage per unit length (or electric field) decreases and consequently decreases the particle velocity as function of deposition time.

In curve D, the deposition rate is lower than other curves for both the contribution of the concentration decrease and of the resistance of deposit layer. As a deduction, the nature of the deposit layer plays an important role on deposition rate.

Fig. 3. Schematic representation of the kinetics of EPD process: curve A, constant currentconstant concentration; curve B, constant current-variable concentration; curve C, constant voltage-constant concentration, curve D, constant voltage-variable concentration.

Adopting the standard scheme of an electrochemical cell, a combined resistive-capacitive model can be used to represent the electrical behaviour of the EPD process (Fig. 4).

In addition to the resistors, discussed above, some capacitors are added to represent the current transient that occurs when the voltage starts to be applied, generally interpreted as due to the establishment of a concentration profile under a diffusion limited regime (Van der Biest & Vandeperre, 1999).

Ferrari (Ferrari et al., 2006) proposed a resistive model for the deposition kinetics, considering a linear relationship between the suspension resistivity and the solid loading in the suspension. This model agrees with the experimental data referred to a long time deposition of yttria stabilized tetragonal zirconia polycrystalline (Y-TZP), that shows a Sshaped variation of the mass per area unit with time.

Recently Baldisserri (Baldisserri et al., 2010) proposed some assumptions to define the electrical behaviour of a potentiostatic EPD cell: a) the effect of capacitive transients on cell current is negligible, b) the faradic resistance of both the electrode/deposit interfaces is

asymptotically with deposition time. Obviously, the decreasing concentration produces a reduction of the final yield and therefore of the rate deposition either at constant current

Comparing curve A and C (both with constant concentration), the final yield is considerably higher in curve A than in curve C. This is due to the electric resistance of the deposit layer, considered higher than that of an equal thickness taken up by the suspension. Therefore, when the process goes on, the voltage per unit length (or electric field) decreases and

In curve D, the deposition rate is lower than other curves for both the contribution of the concentration decrease and of the resistance of deposit layer. As a deduction, the nature of

Fig. 3. Schematic representation of the kinetics of EPD process: curve A, constant currentconstant concentration; curve B, constant current-variable concentration; curve C, constant

Adopting the standard scheme of an electrochemical cell, a combined resistive-capacitive

In addition to the resistors, discussed above, some capacitors are added to represent the current transient that occurs when the voltage starts to be applied, generally interpreted as due to the establishment of a concentration profile under a diffusion limited regime (Van

Ferrari (Ferrari et al., 2006) proposed a resistive model for the deposition kinetics, considering a linear relationship between the suspension resistivity and the solid loading in the suspension. This model agrees with the experimental data referred to a long time deposition of yttria stabilized tetragonal zirconia polycrystalline (Y-TZP), that shows a S-

Recently Baldisserri (Baldisserri et al., 2010) proposed some assumptions to define the electrical behaviour of a potentiostatic EPD cell: a) the effect of capacitive transients on cell current is negligible, b) the faradic resistance of both the electrode/deposit interfaces is

voltage-constant concentration, curve D, constant voltage-variable concentration.

model can be used to represent the electrical behaviour of the EPD process (Fig. 4).

der Biest & Vandeperre, 1999).

shaped variation of the mass per area unit with time.

consequently decreases the particle velocity as function of deposition time.

the deposit layer plays an important role on deposition rate.

(Curve B) or at constant voltage (curve C and D).

constant, c) the electrical resistance of the EPD deposit is proportional to its thickness, d) the electrical resistance of the suspension is constant.

Fig. 4. Electric model of EPD process with plane electrode (we= working electrode, ce= counter electrode, DL= double layer)

The assumption a) is based on the estimate of time required for double-layer charging. The assumption b) is supported by the consideration that after the formation of the first monolayer, the interface at the working electrode/suspension should be stabilised as the charging of double-layer should be completed. Similarly at the interface between counter electrode/suspension, but it is less relevant as no deposition occurs at that electrode.

The assumption c) relies on the compliance of the physical properties of deposited layer comparable with a porous media, where the electrical resistance is proportional to its thickness. The assumption d) is based on the consideration that the deposited layer is less thick than the electrodes distance, therefore the decrease in the volume filled by the suspension is negligible.

Baldisserri applied this model to EPD of TiO2 particles and verified experimentally that the previously mentioned assumptions were satisfied. As a result, they derived a linear correlation between deposited mass and passed charge and a non-linear kinetic equation:

$$m(t) = A \frac{VD\_d}{i\_0 \rho\_d} \left[ (1 + at)^{1/2} - 1 \right], \ a = \frac{2Kl\_o^2}{V} \left( \frac{\rho\_d}{D\_d} \right) \tag{5}$$

Ceramic Coatings Obtained by Electrophoretic Deposition:

(- - -) of an electric field.

௫ୀஶܪ

ା ௧௦௧௦௦

zeta-potential at the cathode and consequently to coagulation.

mechanism when the deposit is thicker than a monolayer.

ሱۛۛۛۛۛۛۛۛۛۛۛۛۛሮܪ௫ୀ

the deposition occurs on a membrane which does not act as an electrode.

the reaction:

particles concentration allows the formation of a particles deposit.

Fundamentals, Models, Post-Deposition Processes and Applications 51

is approaching, it can be close enough to interact through van der Waals attractive forces and so coagulates. Similar mechanism occurs at the electrode surface where the high

Fig. 5. Potential energy vs. separation distance for particles in absence () and in presence

This model was integrated by Fukada (Fukada et al., 2004) with a further consideration based on the experimental observation of De (De & Nicholson, 1999). According to De, Fukada verified that H+ are depleted at the cathode because of particle discharge, through

This implies an increase of local pH toward the isoelectric point and then the coagulation is facilitated. Fukada found an analytical expression for the variation of the H+ concentration with time, validated by experimental results for alumina suspension in ethanol. This expression, generally suitable for suspensions containing H+ or H3O+, shows that the steady state with respect to diffusion and charge transfer of H+ ions corresponds to a reduction of

Other deposition mechanisms, mentioned above, although explain experimental results in some conditions, are not valid in general. In fact, flocculation by particles accumulation suggests a deposit formation by electrophoresis similar to gravitation, so that the pressure exerted by the arriving particles at the electrode makes the particles close to form a deposit, overcoming the repulsion forces between particles. This hypothesis does not explain why

Similarly, particles charge neutralization at the electrode suggests the charged particles are neutralized by the contact with the electrode surface, but it does not explain the deposition

With respect to electrochemical coagulation of particles, the hypothesis is that an increase of electrolyte concentration around particles produces a reduction of the repulsion between particles near the electrode, where particles can coagulate. However, time required to have an increased electrolyte concentration is not negligible, experimentally estimated as

ା ݁ି ௧௦

ሱۛۛۛۛۛۛۛۛۛۛሮ <sup>ଵ</sup>

ଶ

(7) ଶܪ

where *i0*, current density at time *t=0,* K, the deposited mass-passed charge ratio, V, the external applied voltage, d and Dd, resistivity and density of the deposited layer, respectively. This equation at short deposition time is approximated by:

$$m(t) = Ai\_0Kt\tag{6}$$

Therefore, this resistive model is able to describe the EPD process both at regime and during the transient of the deposition current, provided that suspension contains such dispersant or binder as the electroactive chemical species are available for all the deposition time, making deposition a diffusive control process.
