**2.1 Flow diagram of hybrid PZT films**

It is well-accepted that hybrid powder sol gel coating technology is an excellent technique to develop high quality thick ceramic films of more than 1m, while all the benefits of sol gel, i.e. ease of fabrication, ability to coat complex geometries and relative cost effectiveness, remains intact (Barrow et al., 1997; Barrow et al., 1995; Dorey et al., 2002, Pérez et al., 2007). It is also known that high-quality lead zirconate titanate (PZT), yttria- and ceria-stabilized zirconia, titania, silica and alumina thick films with more than 100 m could be fabricated by this method (Barrow et al., 1995). However, questions remain; for instance, which are the advantages and disadvantages of this method?

It is clear that the advantages of the method are associated with the possibility to obtain high-performance thick films with up to 100m on a variety of substrate material and shapes, but what about the disadvantages? The main problem of this method is associated with the difficulty to obtain dense films that reproduce the ceramics properties. This problem affects all type of thick films; however, it is more pronounced in ferroelectric materials like PZT, where the dielectric, ferroelectric and piezoelectric properties are highly influenced by the film densification.

The preparation process of thick films, specifically PZT, is divided into two main steps: i) the selection of the sol-gel matrix and ii) the dispersion of the PZT precursor powder in the sol-gel matrix. The sol-gel matrix is selected taking into account its composition, viscosity, and endurance to aging, etc. Once the selection of the sol-gel matrix is completed, the second step is the dispersion of the PZT powder into PZT sol-gel matrix. The powder dispersion process guarantees the homogeneity of the suspension and eliminates the possibility of agglomerate formation.

Several authors have considered that this step is fundamental in the preparation of high quality thick films (Barrow et al., 1997; Barrow et al., 1995; Pérez et al., 2007; Kholkin et al., 2001; Simon et al., 2001], because it determines the powder agglomeration degree and subsequent densification of the films. Few authors have showed that the level of powder agglomeration inside the sol gel matrix depends on the shape and size and the method used to obtain these powders (Simon et al., 2001). Moreover, it is consensual that agglomeration increases with time, due to the surface tension force and the sedimentation process.

To avoid the agglomeration process that take places before and during the thick film preparation, some authors have used organic dispersants (i.e., buthoxyethoxy-ethyl acetate (BEEA)), high molecular weight solvents (i.e., -terpineol), binders (i.e., polyvinyl butyral (PVB)) and plasticizers (i.e., polyethylene glycol (PEG)). Figure 1 shows the grain size distribution of PZT and PT powders prepared by different synthesis routes, which were dispersed using the above-mentioned compounds (Simon et al., 2001). It is clear that powder preparation processes and consequently grain size and shape have a fundamental role in the powder agglomerate formation. For instances, PZT powder prepared by conventional solid

In this chapter an exhaustive review of the preparation of PZT thick films by infiltration method will be presented. Solution powder agglomeration, film densification, phase formation temperature, among others, will be some of the topics that will be analyzed. Finally, the structural and electrical properties of the PZT thick films as a function of the

It is well-accepted that hybrid powder sol gel coating technology is an excellent technique to develop high quality thick ceramic films of more than 1m, while all the benefits of sol gel, i.e. ease of fabrication, ability to coat complex geometries and relative cost effectiveness, remains intact (Barrow et al., 1997; Barrow et al., 1995; Dorey et al., 2002, Pérez et al., 2007). It is also known that high-quality lead zirconate titanate (PZT), yttria- and ceria-stabilized zirconia, titania, silica and alumina thick films with more than 100 m could be fabricated by this method (Barrow et al., 1995). However, questions remain; for instance, which are the

It is clear that the advantages of the method are associated with the possibility to obtain high-performance thick films with up to 100m on a variety of substrate material and shapes, but what about the disadvantages? The main problem of this method is associated with the difficulty to obtain dense films that reproduce the ceramics properties. This problem affects all type of thick films; however, it is more pronounced in ferroelectric materials like PZT, where the dielectric, ferroelectric and piezoelectric properties are highly

The preparation process of thick films, specifically PZT, is divided into two main steps: i) the selection of the sol-gel matrix and ii) the dispersion of the PZT precursor powder in the sol-gel matrix. The sol-gel matrix is selected taking into account its composition, viscosity, and endurance to aging, etc. Once the selection of the sol-gel matrix is completed, the second step is the dispersion of the PZT powder into PZT sol-gel matrix. The powder dispersion process guarantees the homogeneity of the suspension and eliminates the

Several authors have considered that this step is fundamental in the preparation of high quality thick films (Barrow et al., 1997; Barrow et al., 1995; Pérez et al., 2007; Kholkin et al., 2001; Simon et al., 2001], because it determines the powder agglomeration degree and subsequent densification of the films. Few authors have showed that the level of powder agglomeration inside the sol gel matrix depends on the shape and size and the method used to obtain these powders (Simon et al., 2001). Moreover, it is consensual that agglomeration

To avoid the agglomeration process that take places before and during the thick film preparation, some authors have used organic dispersants (i.e., buthoxyethoxy-ethyl acetate (BEEA)), high molecular weight solvents (i.e., -terpineol), binders (i.e., polyvinyl butyral (PVB)) and plasticizers (i.e., polyethylene glycol (PEG)). Figure 1 shows the grain size distribution of PZT and PT powders prepared by different synthesis routes, which were dispersed using the above-mentioned compounds (Simon et al., 2001). It is clear that powder preparation processes and consequently grain size and shape have a fundamental role in the powder agglomerate formation. For instances, PZT powder prepared by conventional solid

increases with time, due to the surface tension force and the sedimentation process.

**2. Formation and structural characterization of the thick PZT films** 

number of solution infiltrations will be highlighted.

**2.1 Flow diagram of hybrid PZT films** 

advantages and disadvantages of this method?

influenced by the film densification.

possibility of agglomerate formation.

state oxide methods (dry method or DM) shows a mean grain size of 2.2 m, while PZT powder prepared by a coprecipitation process (web method or WM) shows a mean grain size of 1.9 m. Some readers could assume that a progressive decrease in powder size could contribute to a smaller powder agglomeration inside the sol-gel matrix, improving the films densification. However, this is not completely true, as when the power size decreases, the effective surface of the powder increases. This increase in the powder effective surface results in an increase of the surface tension force, which strongly contribute to agglomerate formation. An evident example can be observed in Fig. 1, the coprecipitation process produces smaller grain size PZT powders; however, during the sol-gel PZT power mixture a ~10 m agglomerated is observed.

Fig. 1. Grain size distribution of PZT and PT powders prepared by different synthesis routes, and dispersed using -terpineol solvent, BEEA dispersant, PVB binder and PEG plasticizer (Simon et al., 2001). (Copyright Elsevier)

Similar agglomeration behavior is also observed in high molecular weight free sol gel matrixes. In this case authors have used an ultrasonic dispersion process combined with a relatively high viscous sol-gel matrix in order to eliminate the PZT powder agglomeration (Pérez et al., 2007). As the ultrasonic dispersion time increases, agglomerate formation decreases, as shown in Fig.2. It is clear that the ultrasonic process disperses the agglomerates resulting in an increase of low diameter differential volume. The reduction of the PZT powder agglomeration and the increase in the low diameter differential volume improve the densification of the film during the preparation and reduce the formation of *closed pores*, which cannot be filled by any post-deposition process, such as: infiltration.

It is notable that the ultrasonic process eliminates the agglomerate formation; moreover, it also negates the use of the high molecular weight solvents, dispersants, binders and plasticizer, resulting in shorter drying and pre-annealing processes during the thick film preparation process. The elimination of high molecular weight compounds from the sol-gel matrix reduces in large scale pore formation, improving the PZT thick films densification.

Piezoelectric Thick Films: Preparation and Characterization 355

treatment of each layer should take into account the type of solvent and the organic compounds utilized in the preparation of the PZT sol-gel matrix and also the stoichiometry of the material that is to be prepared. For that reason, prior to the preparation process it is convenient to carry out the decomposition analysis of the sol-gel matrix by using thermogravimetric and differential thermal analysis techniques. These techniques supply the characteristic decomposition temperatures of the sol-gel matrix. Drying and/or preannealing the PZT *green* layers above the characteristic temperatures should be carried out

On the other hand, PZT thick films prepared by a sol gel matrix rich in high molecular weight compounds, like dispersants, binders and plasticizers, show long drying and preannealing processes that range from 1 hour up to 24 hours. The long time drying and preannealing processes are necessary in order to evaporate the solvents and burn all the high molecular weight organic compounds used in the PZT sol-gel matrix preparation avoiding,

Once the desire thickness is obtained the whole film is subjected to a crystallization stage where the pre-annealing sample is sintered at higher temperatures designed to develop the perosvkite phase. In the case of PZT thick films the sintering temperature (also known as annealing temperature) can vary from 400ºC up to 800ºC as a function of the zirconium/titanium ratio, while the annealing time could go from 30 up to 60 minutes. Using this technique, several authors have prepared thick, crack free PZT films (Barrow et al., 1995; Dorey et al., 2002; Pérez et al., 2007). However, Barrow *et. al.,* (Barrow et al., 1995) were the first to attribute the crack free nature of these films to i) the presence of large amounts of powder that results in a decrease of the level of sol-gel present and hence lower shrinkage; (ii) strong bonding between the sol-gel and the PZT particles making cracking

On the other hand, Wu *et. al.,* (Wu et al., 1999) proved that the incorporation of PZT powder into the PZT sol-gel solution shows additional benefits. The addition of ~1 wt.% of PZT powder decreases the perovskite formation temperature by 50 ºC, increasing substantially the dielectric and ferroelectric properties of these films (Wu et al.,1999). It is believed that the incorporation of PZT micro-powders in a PZT sol-gel matrix promotes heterogeneous nucleation of the perovskite phase coming from the sol-gel, resulting in a randomly

Dorey *et. al.*, (Dorey et al., 2002) and Pérez *et. al.*, (Pérez, 2004) observed that a graded structure is obtained when the PZT thick film is not infiltrated with the precursor PZT solgel solution, during the preparation process. This graded structure results because the solgel is drawn from the slurry into the underlying porous composite layer (Dorey et al., 2002). It is believed that as the number of layers increased, the lower composite layers become further enriched with sol-gel. Thus, the bottom composite layer of a four layer structure will be effectively infiltrated three times (Dorey et al., 2002). It is evident that the infiltration of the bottom layer is conditioned by the porous infiltration saturation. Pérez *et. al.*, observed that there is a progressive infiltration of PZT graded structure up to four infiltrations (Pérez, 2004). However, at higher infiltrations the porosity of the bottom layer practically does not change, which can be further explained based on Darcy´s law

It is consensual that the formation of a graded structure is detrimental to the dielectric, ferroelectric and piezoelectric properties of the PZT thick films. Hence, an intermediate

gradually, because it could result in the formation of cracks.

therefore, the formation of crack in the films.

less likely.

orientated PZT film.

(Scheidegger, 1974).

Fig. 2. Distributions of the TRS600 PZT powder particle under different ultrasonic mixing times (Pérez et al., 2007). (Copyright Elsevier)

Hybrid sol-gel/powder formation is followed by the film preparation process. The PZT thick film preparation process practically does not differ to the standard sol-gel thin film preparation (Barrow et al., 1997; Barrow et al., 1995; He et al., 2003; Dorey et al., 2002; Pérez et al., 2007; Kholkin et al., 2001). It is known that PZT thick films could be deposited in several types of substrates; however, it is common that PZT thick films are deposited onto platinized silicon wafers (Pt/Ti/SiO2/Si), due to the higher conductivity of the platinum (Barrow et al., 1995). Prior to the coating step the substrate should be cleaned to remove the substrate dirt. It is usual that the removing of any residual organics of the substrate surface involves low molecular weight volatile compounds, such us: methanol, ethanol, isopropanol, and acetone, among others. However, other techniques like ultrasonic bath and plasma etching are regularly used during the substrate cleaning process (Dorey et al., 2002; Pérez et al., 2007). Afterward, PZT thick films are grown up by depositing a consecutive number of layers. Each layer consists of an initial composite layer, which is deposited by covering the entire wafer surface with the composite slurry and then spinning for 30 s to 60 s in the 2000-4000 rpm spinning frequency range. The spinning frequency and time are normally optimized in order to obtain the desire thickness of each individual layer, which depends on the powder mean size and also on the viscosity of the sol-gel matrix.

Each individual layer is then subjected to a heat treatment process at an intermediate temperature designed to remove the organic component and to pyrolise the PZT sol-gel. This heat treatment process is divided in two step: 1) the drying process, which is carried out from 100ºC up to 300ºC during ~60 s and 2) the pre-annealing process (or calcination) that is carried in the 300ºC-500ºC temperature range form few seconds up to various minutes (Barrow et al., 1997; Barrow et al., 1995; He et al., 2003; Dorey et al., 2002; Pérez et al., 2007; Kholkin et al., 2001). The drying and pre-annealing temperature used in the heat

 0 min 5 min 10 min 20 min 30 min

0.01 0.1 1 10 100

diameter (m)

Hybrid sol-gel/powder formation is followed by the film preparation process. The PZT thick film preparation process practically does not differ to the standard sol-gel thin film preparation (Barrow et al., 1997; Barrow et al., 1995; He et al., 2003; Dorey et al., 2002; Pérez et al., 2007; Kholkin et al., 2001). It is known that PZT thick films could be deposited in several types of substrates; however, it is common that PZT thick films are deposited onto platinized silicon wafers (Pt/Ti/SiO2/Si), due to the higher conductivity of the platinum (Barrow et al., 1995). Prior to the coating step the substrate should be cleaned to remove the substrate dirt. It is usual that the removing of any residual organics of the substrate surface involves low molecular weight volatile compounds, such us: methanol, ethanol, isopropanol, and acetone, among others. However, other techniques like ultrasonic bath and plasma etching are regularly used during the substrate cleaning process (Dorey et al., 2002; Pérez et al., 2007). Afterward, PZT thick films are grown up by depositing a consecutive number of layers. Each layer consists of an initial composite layer, which is deposited by covering the entire wafer surface with the composite slurry and then spinning for 30 s to 60 s in the 2000-4000 rpm spinning frequency range. The spinning frequency and time are normally optimized in order to obtain the desire thickness of each individual layer, which

Fig. 2. Distributions of the TRS600 PZT powder particle under different ultrasonic mixing

depends on the powder mean size and also on the viscosity of the sol-gel matrix.

Each individual layer is then subjected to a heat treatment process at an intermediate temperature designed to remove the organic component and to pyrolise the PZT sol-gel. This heat treatment process is divided in two step: 1) the drying process, which is carried out from 100ºC up to 300ºC during ~60 s and 2) the pre-annealing process (or calcination) that is carried in the 300ºC-500ºC temperature range form few seconds up to various minutes (Barrow et al., 1997; Barrow et al., 1995; He et al., 2003; Dorey et al., 2002; Pérez et al., 2007; Kholkin et al., 2001). The drying and pre-annealing temperature used in the heat

0

times (Pérez et al., 2007). (Copyright Elsevier)

1

2

3

differential volume (%)

4

5

6

treatment of each layer should take into account the type of solvent and the organic compounds utilized in the preparation of the PZT sol-gel matrix and also the stoichiometry of the material that is to be prepared. For that reason, prior to the preparation process it is convenient to carry out the decomposition analysis of the sol-gel matrix by using thermogravimetric and differential thermal analysis techniques. These techniques supply the characteristic decomposition temperatures of the sol-gel matrix. Drying and/or preannealing the PZT *green* layers above the characteristic temperatures should be carried out gradually, because it could result in the formation of cracks.

On the other hand, PZT thick films prepared by a sol gel matrix rich in high molecular weight compounds, like dispersants, binders and plasticizers, show long drying and preannealing processes that range from 1 hour up to 24 hours. The long time drying and preannealing processes are necessary in order to evaporate the solvents and burn all the high molecular weight organic compounds used in the PZT sol-gel matrix preparation avoiding, therefore, the formation of crack in the films.

Once the desire thickness is obtained the whole film is subjected to a crystallization stage where the pre-annealing sample is sintered at higher temperatures designed to develop the perosvkite phase. In the case of PZT thick films the sintering temperature (also known as annealing temperature) can vary from 400ºC up to 800ºC as a function of the zirconium/titanium ratio, while the annealing time could go from 30 up to 60 minutes. Using this technique, several authors have prepared thick, crack free PZT films (Barrow et al., 1995; Dorey et al., 2002; Pérez et al., 2007). However, Barrow *et. al.,* (Barrow et al., 1995) were the first to attribute the crack free nature of these films to i) the presence of large amounts of powder that results in a decrease of the level of sol-gel present and hence lower shrinkage; (ii) strong bonding between the sol-gel and the PZT particles making cracking less likely.

On the other hand, Wu *et. al.,* (Wu et al., 1999) proved that the incorporation of PZT powder into the PZT sol-gel solution shows additional benefits. The addition of ~1 wt.% of PZT powder decreases the perovskite formation temperature by 50 ºC, increasing substantially the dielectric and ferroelectric properties of these films (Wu et al.,1999). It is believed that the incorporation of PZT micro-powders in a PZT sol-gel matrix promotes heterogeneous nucleation of the perovskite phase coming from the sol-gel, resulting in a randomly orientated PZT film.

Dorey *et. al.*, (Dorey et al., 2002) and Pérez *et. al.*, (Pérez, 2004) observed that a graded structure is obtained when the PZT thick film is not infiltrated with the precursor PZT solgel solution, during the preparation process. This graded structure results because the solgel is drawn from the slurry into the underlying porous composite layer (Dorey et al., 2002). It is believed that as the number of layers increased, the lower composite layers become further enriched with sol-gel. Thus, the bottom composite layer of a four layer structure will be effectively infiltrated three times (Dorey et al., 2002). It is evident that the infiltration of the bottom layer is conditioned by the porous infiltration saturation. Pérez *et. al.*, observed that there is a progressive infiltration of PZT graded structure up to four infiltrations (Pérez, 2004). However, at higher infiltrations the porosity of the bottom layer practically does not change, which can be further explained based on Darcy´s law (Scheidegger, 1974).

It is consensual that the formation of a graded structure is detrimental to the dielectric, ferroelectric and piezoelectric properties of the PZT thick films. Hence, an intermediate

Piezoelectric Thick Films: Preparation and Characterization 357

equation is accurate when the pressure gradient in the liquid column is balanced by gravitational forces. For this equilibrium state, the Darcy velocity is zero. The equations propose that for a "small" imbalance in the gradients that drive the flow, the Darcy expressions reasonably describe the velocity (or infiltration velocity). Confirming experimental evidence indicates that Darcy's correlation is a useful expression to describe flow through porous medium (Scheidegger, 1974). It is clear that the Darcy velocity is not a true velocity of the fluid but represents an effective flow rate through the porous medium. Several pressures could influence the infiltration process. However, it can be resumed to three main pressures, the applied pressure *Pa* (which includes the hydrostatic pressure), the capillarity force *Pc* and the internal opposing pressure of the compressed gas *Pi*, as shown by Kholkin et. al., (Kholkin et al., 2001). During a "static" infiltration, such as used during a dip-coating process, the infiltration is mainly dominated by the hydrostatic pressure, the capillarity pressure and the internal opposing pressure of the compressed gas; however, during a "dynamic" infiltration, such as used during a spin-coating process, the infiltration

( ) 

is the density of the liquid and *P* the pressure experimented by the liquid. This

(4)

(5)

(6)

*K g P g <sup>l</sup> l* 

*q*

is mainly dominated by the applied pressure due to the centrifugation process.

*z*

permeability of a powder compact is expressed by the Kozeny-Carman expression:

*<sup>K</sup> <sup>D</sup>*

*<sup>C</sup>* 2 3 2 1 36 ( ) 

where *D* is the average grain size and *C* de is a parameter that define the shape and tortuosity of the porous channel (Scheidegger, 1974). The *C* value for a powder compact is estimated as ≈5; however, it increases exponentially as the number of infiltrations increase, mainly because the porous size is reduced. It is believed that this is the major factor that reduces the infiltration depth in powder compact in general and in PZT thick films in

Kholkin et al., (Kholkin et al., 2001) has predicted based on the modified Darcy´s law reported by Scheidegger et. al., (Scheidegger, 1974) (see Equation 5) that for a static infiltration process, where the flow of the liquid into the porous media in controlled by the capillarity pressure, the distance of liquid introduced in the PZT thick film (*z*) during the

> *KP* ( ) *<sup>t</sup>* 2 <sup>1</sup> 2 1 2

This value is 200 times greater than the thickness of the PZT thick film deposited in this study (~10nm). Authors suggest that this value was reached because they did not take into account the increase of the internal opposing pressure of the compressed gas due to the no evacuation of the replaced gas (Kholkin et al., 2001). However, there are other factors like the viscosity of the solution and the permeability of the porous body that should be taken into account. It is known that as the number of infiltrations increase, the viscosity of the solution increases, resulting in a decrease of the infiltration depth (Dorey et al., 2002). On the other hand, the

where 

time (*t*=30s) is ≈ 2mm.

particular.

infiltration of each individual composite layer and a final infiltration of the whole film are necessary, leading to a homogenization of the film structure, fixing the graded structural problem, as shown by Perez and co-workers (Pérez et al., 2007; Pérez, 2004). The infiltration of each individual layer with sol-gel prior to the deposition of the next composite layer results in a relative densification of the layer and a strengthening of the powder compact without shrinkage, while the final infiltration is used to improve the PZT thick film surface (Tu et al., 1995). Thus, with this method it is possible to produce high-quality PZT thick composite films with uniform densities and good surfaces.

#### **2.2 Infiltration of the PZT thick films**

It was mentioned above that the infiltration process of the PZT thick film could be explained based on Darcy´s law (Scheidegger, 1974). The experiments of Darcy were focused on the volumetric flow rate (Q) of the fluid through a sand column, which is similar to the infiltration of a sol-gel solution through a porous film. It can be observed that flow occurs only in the pore space. Thus, the effective area of flow is not the entire column cross section (A), but this area multiplied by the porosity (A). Note that although the porosity is a volume fraction, it is also useful in determining an average effective area of flow. Thus the average speed of the macroscopically one-dimensional flow in a cross section of Darcy's column relative to the solid grains is:

$$\left|\mathbf{v}^{l} - \mathbf{v}^{s}\right| = \frac{\underline{Q}}{\mu A} \tag{1}$$

where *v<sup>l</sup>* is the velocity of the liquid in a porous media and *v<sup>s</sup>* is the velocity of the media. It is clear that in the thick films case the velocity of the media is ~0.

Taking into account that Darcy demonstrated experimentally that the volumetric flow rate of a liquid (i.e., water) down through the porous medium is proportional to the head difference across the sand column (*h2 – h1*), and the cross sectional flow area and is inversely proportional to the packed height of the column (L), such that:

$$
\underline{Q} = KA \left| \frac{h\_2 - h\_1}{L} \right| \tag{2}
$$

where *K* is referred to as hydraulic conductivity (or permeability of the porous body) and it is a function of both the porous medium and the fluid properties, being practically constant for a particular packing even when the flow rate in the column changes. One can obtain the differential form of Darcy´s law, combining the equations 1 and 2 when the limit of the column length is reduce (L0), as shown:

$$
\mu(\mathbf{v}^l - \mathbf{v}^s) = K^l \frac{dh}{dz} \tag{3}
$$

where the hydraulic conductivity (*Kl* ) has been labelled with a superscript (*l* ) to emphasize that in a particular medium, its value will depend on the properties of the fluid phase. *q vv l ls* ( )In a general case the Darcy velocity can be represented in terms of the gradient of pressure elevation heads as:

infiltration of each individual composite layer and a final infiltration of the whole film are necessary, leading to a homogenization of the film structure, fixing the graded structural problem, as shown by Perez and co-workers (Pérez et al., 2007; Pérez, 2004). The infiltration of each individual layer with sol-gel prior to the deposition of the next composite layer results in a relative densification of the layer and a strengthening of the powder compact without shrinkage, while the final infiltration is used to improve the PZT thick film surface (Tu et al., 1995). Thus, with this method it is possible to produce high-quality PZT thick

It was mentioned above that the infiltration process of the PZT thick film could be explained based on Darcy´s law (Scheidegger, 1974). The experiments of Darcy were focused on the volumetric flow rate (Q) of the fluid through a sand column, which is similar to the infiltration of a sol-gel solution through a porous film. It can be observed that flow occurs only in the pore space. Thus, the effective area of flow is not the entire column cross section

volume fraction, it is also useful in determining an average effective area of flow. Thus the average speed of the macroscopically one-dimensional flow in a cross section of Darcy's

*v v <sup>Q</sup>*

Taking into account that Darcy demonstrated experimentally that the volumetric flow rate of a liquid (i.e., water) down through the porous medium is proportional to the head difference across the sand column (*h2 – h1*), and the cross sectional flow area and is inversely

*Q KA h h*

where *K* is referred to as hydraulic conductivity (or permeability of the porous body) and it is a function of both the porous medium and the fluid properties, being practically constant for a particular packing even when the flow rate in the column changes. One can obtain the differential form of Darcy´s law, combining the equations 1 and 2 when the limit of the

( ) *vv K dh*

( )In a general case the Darcy velocity can be represented in terms of the gradient of

*ls l*

that in a particular medium, its value will depend on the properties of the fluid phase.

*dz*

*l s*

is the velocity of the liquid in a porous media and *v<sup>s</sup>*

It is clear that in the thick films case the velocity of the media is ~0.

proportional to the packed height of the column (L), such that:

*A*

A). Note that although the porosity

*<sup>L</sup>* 2 1 (2)

) has been labelled with a superscript (*l* ) to emphasize

is a

is the velocity of the media.

(1)

(3)

composite films with uniform densities and good surfaces.

**2.2 Infiltration of the PZT thick films** 

column relative to the solid grains is:

column length is reduce (L0), as shown:

where the hydraulic conductivity (*Kl*

pressure elevation heads as:

where *v<sup>l</sup>*

*q vv l ls* 

(A), but this area multiplied by the porosity (

$$q^l = \frac{K^l}{\rho \mathbf{g}} (\nabla P - \rho \mathbf{g}) \tag{4}$$

where is the density of the liquid and *P* the pressure experimented by the liquid. This equation is accurate when the pressure gradient in the liquid column is balanced by gravitational forces. For this equilibrium state, the Darcy velocity is zero. The equations propose that for a "small" imbalance in the gradients that drive the flow, the Darcy expressions reasonably describe the velocity (or infiltration velocity). Confirming experimental evidence indicates that Darcy's correlation is a useful expression to describe flow through porous medium (Scheidegger, 1974). It is clear that the Darcy velocity is not a true velocity of the fluid but represents an effective flow rate through the porous medium.

Several pressures could influence the infiltration process. However, it can be resumed to three main pressures, the applied pressure *Pa* (which includes the hydrostatic pressure), the capillarity force *Pc* and the internal opposing pressure of the compressed gas *Pi*, as shown by Kholkin et. al., (Kholkin et al., 2001). During a "static" infiltration, such as used during a dip-coating process, the infiltration is mainly dominated by the hydrostatic pressure, the capillarity pressure and the internal opposing pressure of the compressed gas; however, during a "dynamic" infiltration, such as used during a spin-coating process, the infiltration is mainly dominated by the applied pressure due to the centrifugation process.

Kholkin et al., (Kholkin et al., 2001) has predicted based on the modified Darcy´s law reported by Scheidegger et. al., (Scheidegger, 1974) (see Equation 5) that for a static infiltration process, where the flow of the liquid into the porous media in controlled by the capillarity pressure, the distance of liquid introduced in the PZT thick film (*z*) during the time (*t*=30s) is ≈ 2mm.

$$z = (\frac{2KP}{\eta})^{\frac{1}{2}} t^{\frac{1}{2}} \tag{5}$$

This value is 200 times greater than the thickness of the PZT thick film deposited in this study (~10nm). Authors suggest that this value was reached because they did not take into account the increase of the internal opposing pressure of the compressed gas due to the no evacuation of the replaced gas (Kholkin et al., 2001). However, there are other factors like the viscosity of the solution and the permeability of the porous body that should be taken into account. It is known that as the number of infiltrations increase, the viscosity of the solution increases, resulting in a decrease of the infiltration depth (Dorey et al., 2002). On the other hand, the permeability of a powder compact is expressed by the Kozeny-Carman expression:

$$K = \frac{D^2(1-\rho)^3}{36C\rho^2} \tag{6}$$

where *D* is the average grain size and *C* de is a parameter that define the shape and tortuosity of the porous channel (Scheidegger, 1974). The *C* value for a powder compact is estimated as ≈5; however, it increases exponentially as the number of infiltrations increase, mainly because the porous size is reduced. It is believed that this is the major factor that reduces the infiltration depth in powder compact in general and in PZT thick films in particular.

Piezoelectric Thick Films: Preparation and Characterization 359

PZT TRS600

(112)

(211)

(022)

PZT (110) peak (220)

(212)

(003)

(221)

(300)

20 30 40 50 60 70 80

2 Thetaº

26 27 28 29 30 31 32 33 34

2 Thetaº

Fig. 3. X-ray diffraction patterns of a) TRS600 PZT precursor powder and b) PZT thick films prepared with different number of top infiltrations (0, 2, 4, 6, and 8) (Pérez et al., 2007).

Finally, a decrease in the intensity of the extra TRS600 X-ray diffraction peaks (visible at 26.8o and 33.12o) is observed, showing that as the number of infiltrations increase the factional volume of the formed sol-gel PZT phase is more palpable. This fact emphasizes the idea that as the number of infiltrations increases the number of pores decreases due to a complete coverage by the sol-gel solution. Moreover, the decrease in the 26.8o X-ray diffraction peak could be also associated with the possible evaporation of the lead oxide

(102) (210)

(201)

(200)

(002)

(111)

(110)

(100)

(001)

Intensity (a.u)

Intensity (a.u)

(Copyright Elsevier)

PbO (Powder peak)

during the PZT thick film heat treatment process.

(310)

a)

(301)

(Powder peak)

b)

(103)

In the next section, one analyzes the effect of the number of top infiltrations in structural, dielectric and piezoelectric properties of a intermediate infiltrated PZT thick film. Moreover, the dielectric properties of the infiltrated PZT thick films will be simulated based on 0-3 and cube ceramic/ceramic composite models, where the numbers 0 and 3 describe the connectivity of the two phases of the material (i.e., the sol gel matrix interconnected in the three directions (3) whereas PZT powder particles are not connected in any direction (0)) (Newnham et al., 1978). Finally, the structural and electrical results will be compared with the ones reported by Dorey *et. al.,* and Ohno *et al.,* in PZT thick film infiltrated with a high molecular weight prepared PZT solution (Ohno et al., 2000; Pérez et al., 2007).
