**Spray-Drying vs. freeze drying**

The two main drying techniques for obtaining ready to press powders are spray drying and freeze drying. In spray drying process, the solvent is eliminated by evaporation when the slurry is passed through a chamber at a temperature over the solvent boiling point, whereas during freeze drying the suspension is previously frozen and then water is removed by sublimation. The two processes are similar, except for energy flow. In the case of spray drying, energy is applied to the droplet, forcing evaporation of the medium resulting in both energy and mass transfer through the droplet. In spray freeze drying, energy only is removed from the droplet, forcing the melted to solidify. Both techniques are schematized in figure 3.2.

Spray drying is the most widely used industrial process involving particle formation and drying. It is highly suited for the continuous production of dry solids in either powder, granulate or agglomerate form from liquid feedstocks as solutions, emulsions and pumpable suspensions.

There are three fundamental steps involved in spray drying.


Spray drying involves the atomization of a liquid feedstock into a spray of droplets and contacting the droplets with hot air in a drying chamber.

However, slip casting has limitations, which, from an operational point of view, are primarily its slow casting rates, with thickness proportional to the square root of casting time, and hence increased cost of casting and of drying to avoid cracking, as well as costs for preparing and maintaining a large mould inventory and facilities for mould storage and drying. Using pressure to accelerate the dewatering process improves the productivity of the process. The main difference in comparison with slip casting is that the water is not removed by capillary suction (negative pressure in the plaster mould) but by pressurisation (positive slip pressure). Control of the filtration process is based on four parameters; the pressure differential on the body, the liquid-medium viscosity, the specific surface area of the slip's solids content and the body porosity (body formation is dependent on the

In order to solve those problems a great diversity of advanced forming techniques has been developed. Amongst them, aqueous injection moulding, centrifugal slip casting, direct coagulation casting, electrophoretic casting, gelcasting, hydrolysis assisted solidifications, etc can be mentioned. In all cases, the objective is to obtain a very homogenous green body

However, in other cases, stable suspensions of ceramic powders are prepared in order to produce dry powders especially conditioned for forming by pressing. In this case, the most critical issue is to avoid the formation of hard agglomerates that could lead to formation of defects in the final material which could not be removed during firing. Two drying

The two main drying techniques for obtaining ready to press powders are spray drying and freeze drying. In spray drying process, the solvent is eliminated by evaporation when the slurry is passed through a chamber at a temperature over the solvent boiling point, whereas during freeze drying the suspension is previously frozen and then water is removed by sublimation. The two processes are similar, except for energy flow. In the case of spray drying, energy is applied to the droplet, forcing evaporation of the medium resulting in both energy and mass transfer through the droplet. In spray freeze drying, energy only is removed from the droplet, forcing the melted to solidify. Both techniques are schematized in

Spray drying is the most widely used industrial process involving particle formation and drying. It is highly suited for the continuous production of dry solids in either powder, granulate or agglomerate form from liquid feedstocks as solutions, emulsions and pumpable

2. Mixing of these spray droplets with a heated gas stream, allowing the liquid to

Spray drying involves the atomization of a liquid feedstock into a spray of droplets and

techniques are especially suitable for this purpose; spray drying and freeze-drying.

permeability of the layer of body that has already formed from filtered material).

in order to facilitate the preparation of defect free materials.

There are three fundamental steps involved in spray drying.

contacting the droplets with hot air in a drying chamber.

3. Dried powder is separated from the gas stream and collected.

1. Atomization of a liquid feed into fine droplets.

evaporate and leave dried solids.

**Spray-Drying vs. freeze drying** 

figure 3.2.

suspensions.

Fig. 3.2. Spray-drying and freeze–drying processes.

The sprays are produced by either rotary (wheel) or nozzle atomizers. Evaporation of moisture from the droplets and formation of dry particles proceed under controlled temperature and airflow conditions. Powder is discharged continuously from the drying chamber. Operating conditions and dryer design are selected according to the drying characteristics of the product and powder specification.

Spray formation is usually coupled to freeze drying process. This technique is named spray freeze drying. This process consists of


The advantage of using conditioned powder for obtaining transparent ceramic materials is known (I. Amato et al., 1976). The enhanced behaviour during compaction of spray dried or freeze dried powders leads to a more homogeneous distribution of particles in the green body and finally a reduction in the residual porosity of the material. Nevertheless, the incorporation of additives during slurry preparation or binders for favouring soft granulation of powders makes an additional process before firing necessary. This process is named debinding.

Binders, which are used in the slip casting process or in pressing, give the green body a sufficient strength by gluing together particles at their boundary surfaces. Usually those binders are based on polyvinyl alcohols (PVA), polyacrylate or cellulose. High-polymeric compounds such as cellulose and polysaccharides work as plastification agents. They make the flow of ceramic masses during extruding possible.

The thermal treatment of the debinding process destroys the polymers by oxidation or combustion in oxygen containing atmosphere. Very often it is an uncontrolled reaction of

Sintering to Transparency of Polycrystalline Ceramic Materials 539

Fig. 3.4. Schematic representation of phenomena involved during firing of ceramics.

light scattering.

al., 2008).

When the objective is to obtain transparent materials, it is necessary to control the microstructure evolution during ceramic sintering. The grain growth that occurs during sintering reduces the optical transmittance due to two reasons. Firstly, the pore size distribution in the sintered material is proportional to average grain size. Secondly, in the case of birefringent materials as alumina, the higher the grain size the more important the

Grain growth inhibition can be attained by different methods. The most common strategies for reducing grain growth during sintering are the design of complex sintering cycles as two step sintering or the use of dopants for blocking coarsening mechanisms. Two-step sintering has been reported for preparing nanograin ceramics. In the process, the sample is first heated to a higher temperature to achieve an intermediate relative density (above 75%), and then rapidly cooled down and held at a lower temperature until it is fully dense (Z. Chen et

On the other hand, additives can impede the grain growth of ceramics during sintering. In particular, there are many studies treating the alumina grain growth inhibition using different dopants (S.J. Bennison et al., 1990a) (I. Alvarez et.al, 2010). An innovative method

high reaction rate inside the shaped part creating a high gas pressure, which can lead to ruptures of the compact. Debinding process is schematized in figure 3.3.

Fig. 3.3. Description of debinding process.

Debinding process is a critical step. In order to overcome the problems in thermal debinding, solvent debinding has been widely adopted by industry. In this process, a portion of the binder can be chemically removed by using solvents like acetone, trichloroethane or heptane. A large amount of open porosities, after solvent debinding, allows the degraded products to diffuse to the surface easily. A more environmentally friendly method is given by binder compositions containing water-soluble components, like polyethylene glycol.

#### **Sintering**

Finally, the green body is sintered by heating at high temperature in order to eliminate porosity and obtain the desired microstructure. The driving force for sintering is the reduction in surface free energy of the consolidated mass of particles. This reduction in energy can be accomplished by atom diffusion processes that lead to either densification of the body (by transport matter from inside the grains into the pores) or coarsening of the microstructure (by rearrangement of matter between different parts of the pore surfaces without actually leading to a decrease in the pore volume). The diffusion paths for densification and coarsening are shown in figure 3.4 for an idealized situation of two spherical particles in contact. The densification processes remove material from the grain boundary region leading to shrinkage whereas coarsening processes produce microstructural changes without causing shrinkage.

high reaction rate inside the shaped part creating a high gas pressure, which can lead to

Debinding process is a critical step. In order to overcome the problems in thermal debinding, solvent debinding has been widely adopted by industry. In this process, a portion of the binder can be chemically removed by using solvents like acetone, trichloroethane or heptane. A large amount of open porosities, after solvent debinding, allows the degraded products to diffuse to the surface easily. A more environmentally friendly method is given by binder compositions containing water-soluble components, like

Finally, the green body is sintered by heating at high temperature in order to eliminate porosity and obtain the desired microstructure. The driving force for sintering is the reduction in surface free energy of the consolidated mass of particles. This reduction in energy can be accomplished by atom diffusion processes that lead to either densification of the body (by transport matter from inside the grains into the pores) or coarsening of the microstructure (by rearrangement of matter between different parts of the pore surfaces without actually leading to a decrease in the pore volume). The diffusion paths for densification and coarsening are shown in figure 3.4 for an idealized situation of two spherical particles in contact. The densification processes remove material from the grain boundary region leading to shrinkage whereas coarsening processes produce

ruptures of the compact. Debinding process is schematized in figure 3.3.

Fig. 3.3. Description of debinding process.

microstructural changes without causing shrinkage.

polyethylene glycol.

**Sintering** 

Fig. 3.4. Schematic representation of phenomena involved during firing of ceramics.

When the objective is to obtain transparent materials, it is necessary to control the microstructure evolution during ceramic sintering. The grain growth that occurs during sintering reduces the optical transmittance due to two reasons. Firstly, the pore size distribution in the sintered material is proportional to average grain size. Secondly, in the case of birefringent materials as alumina, the higher the grain size the more important the light scattering.

Grain growth inhibition can be attained by different methods. The most common strategies for reducing grain growth during sintering are the design of complex sintering cycles as two step sintering or the use of dopants for blocking coarsening mechanisms. Two-step sintering has been reported for preparing nanograin ceramics. In the process, the sample is first heated to a higher temperature to achieve an intermediate relative density (above 75%), and then rapidly cooled down and held at a lower temperature until it is fully dense (Z. Chen et al., 2008).

On the other hand, additives can impede the grain growth of ceramics during sintering. In particular, there are many studies treating the alumina grain growth inhibition using different dopants (S.J. Bennison et al., 1990a) (I. Alvarez et.al, 2010). An innovative method

Sintering to Transparency of Polycrystalline Ceramic Materials 541

0 20 40 60 260 280 300 320 340

**Time (minutes)**

Fig. 3.6. Comparison of cycle duration between Hot-Pressing and Spark Plasma Sintering

Hot press

Fig. 3.5. Description of SPS process.

0

400

800

**Temperature (oC)**

1200

1600

2000

SPS

for controlling alumina grain growth is described by M. Suárez et al., (2011). In this case, the dopant used is an alumina precursor in order to obtain pure alumina after sintering. The effect of alumina precursor doping on process kinetics and microstructure evolution is discussed.

However, the residual porosity of materials after conventional sintering is usually high enough to reduce noticeably their transmittance or even lead to opaque materials. Thus, an additional process for removing that porosity is usually needed. An effective method for attaining complete densification is the post-HIP treatment. In hot isostatic pressing, the sample predensified, until only closed porosity is present, is placed in a pressure vessel. The pressurizing gas is introduced by means of a compressor to achieve a given initial gas pressure, and the sample is heated to the sintering temperature. During this time the gas pressure increases further to the required value and collapses around the sample, thereby acting to transmit the isostatic pressure to the sample. The use of this technique for improving transparency has been shown by different authors (K. Itatani et al., 2006 and M. Suárez et al., 2010).

#### **Pressure assisted sintering techniques**

The simultaneous application of pressure and heat is also used in the Hot-Pressing and Spark Plasma Sintering, methods that are known as pressure assisted sintering techniques. In this case, a mechanical uniaxial pressure is applied to the sample placed in a dye by a vertical piston while the system is heated. Graphite is the most common die material because it is relatively inexpensive, is easily machined, and has excellent creep resistance at high temperatures. The rate of densification can be deduced by following the piston displacement. Typically, in Hot- Pressing, the sintering temperature is chosen to achieve full densification in 30 min to 2 h. Some guidance for selecting the appropriate hot pressing temperature may be obtained from pressure sintering maps, but trial and error is usually done. Pressure is often maintained during the cooldown step.

A sintering method with a configuration similar to Hot-Pressing has been recently developed. It is named Spark Plasma Sintering (SPS) and its main characteristic is that a pulsed DC current is directly passed through the graphite die while uniaxial pressure is applied. The characteristics include (a) high heating rate, (b) the application of a pressure, and (c) the effect of the current. The main advantage in comparison with other sintering techniques is the high heating rates that can be applied during sample sintering. The description of SPS process and a comparison of cycle duration with Hot Pressing are shown in figures 3.5 and 3.6

These features have a great influence on the characteristics of the materials obtained. The extremely short processing times allow obtaining special microstructures in the final material that are unattainable by other sintering techniques. Thus, it is possible to fabricate a dense material with an average grain size similar to starting powders. There are many scientific works showing the potential of this technique for obtaining a wide diversity of transparent polycrystalline materials (B. N. Kim et al., 2007), (C. Wang et al., 2009), (G. Zhang et al., 2009), (R. Chaim et al., 2006). Nowadays, one of the challenges related with this technique is the scaling-up in order to obtain large samples or complex shape components.

for controlling alumina grain growth is described by M. Suárez et al., (2011). In this case, the dopant used is an alumina precursor in order to obtain pure alumina after sintering. The effect of alumina precursor doping on process kinetics and microstructure evolution is

However, the residual porosity of materials after conventional sintering is usually high enough to reduce noticeably their transmittance or even lead to opaque materials. Thus, an additional process for removing that porosity is usually needed. An effective method for attaining complete densification is the post-HIP treatment. In hot isostatic pressing, the sample predensified, until only closed porosity is present, is placed in a pressure vessel. The pressurizing gas is introduced by means of a compressor to achieve a given initial gas pressure, and the sample is heated to the sintering temperature. During this time the gas pressure increases further to the required value and collapses around the sample, thereby acting to transmit the isostatic pressure to the sample. The use of this technique for improving transparency has been shown by different authors (K. Itatani et al., 2006 and M.

The simultaneous application of pressure and heat is also used in the Hot-Pressing and Spark Plasma Sintering, methods that are known as pressure assisted sintering techniques. In this case, a mechanical uniaxial pressure is applied to the sample placed in a dye by a vertical piston while the system is heated. Graphite is the most common die material because it is relatively inexpensive, is easily machined, and has excellent creep resistance at high temperatures. The rate of densification can be deduced by following the piston displacement. Typically, in Hot- Pressing, the sintering temperature is chosen to achieve full densification in 30 min to 2 h. Some guidance for selecting the appropriate hot pressing temperature may be obtained from pressure sintering maps, but trial and error is usually

A sintering method with a configuration similar to Hot-Pressing has been recently developed. It is named Spark Plasma Sintering (SPS) and its main characteristic is that a pulsed DC current is directly passed through the graphite die while uniaxial pressure is applied. The characteristics include (a) high heating rate, (b) the application of a pressure, and (c) the effect of the current. The main advantage in comparison with other sintering techniques is the high heating rates that can be applied during sample sintering. The description of SPS process and a comparison of cycle duration with Hot Pressing are shown

These features have a great influence on the characteristics of the materials obtained. The extremely short processing times allow obtaining special microstructures in the final material that are unattainable by other sintering techniques. Thus, it is possible to fabricate a dense material with an average grain size similar to starting powders. There are many scientific works showing the potential of this technique for obtaining a wide diversity of transparent polycrystalline materials (B. N. Kim et al., 2007), (C. Wang et al., 2009), (G. Zhang et al., 2009), (R. Chaim et al., 2006). Nowadays, one of the challenges related with this technique is the scaling-up in order to obtain large samples or complex

discussed.

Suárez et al., 2010).

in figures 3.5 and 3.6

shape components.

**Pressure assisted sintering techniques** 

done. Pressure is often maintained during the cooldown step.

Fig. 3.5. Description of SPS process.

Fig. 3.6. Comparison of cycle duration between Hot-Pressing and Spark Plasma Sintering

Sintering to Transparency of Polycrystalline Ceramic Materials 543

non-cubic materials, of which alumina is the most intensively studied material, there is an additional scattering given by the dependence of the value of the refractive index with the orientation of the crystalline grain, which we shall refer to hereafter as scattering due to

Whenever a light beam reaches a surface separating two media with different refractive

n1·sinθ1 = n2·sinθ2 (1)

θ1

n1

n2

θ2

Fig. 4.3. Light refraction at (a) a matrix-pore boundary and (b) between two grains with

θ2

A model based on geometrical optics was first produced by A. L. Dalisa and R. J. Seymour in which a spread function describing the effect on an incident collimated beam of randomly oriented spatial regions with a difference in the refractive index between zero and ± Δnmax

different crystallographic orientations in a birefringent polycrystalline material.

Fig. 4.2. Light scattering sources in polycrystalline ceramics.

indices (see Figure 4.3), it deviates according to the Snell's law:

a) b)

n1 n2

θ1

grains.

#### **4. Characterization of transparent materials**

Transparency represents the ability of materials to allow the transmission of light through them. When traversing a transparent material, a light beam propagates along the same direction before and after traversing the material (see Fig. 4.1a). However, perfectly transparent materials are rare in nature and most of them present the so-called scattering centres. When the concentration of scattering centres is sufficiently large, the material still allows the transmission of light, but the transmitted light beam does not only propagate along the incident direction, but diffuse, off-scattered, beams can be detected; in this case the material becomes translucent (see Fig. 4.1b). In this work, we shall refer only to the transparency of a material, which is measured as the real in-line transmittance. When the amount of scattering centres is higher, the materials scatter so much light that they become opaque (see Fig. 4.1c).

Fig. 4.1. Light propagation through a transparent (a), a translucent (b) and a highly dispersive, almost opaque (c) material.

Therefore, it becomes necessary to define a measurement technique that allows distinguishing between translucency and transparency. In this case, the relevant parameter is the real in line transmittance (Appetz & Bruggen, 2003) or RIT. In this case, a collimated light beam impinges on the sample and the detector is placed far away, usually 1 m, from the sample. This way, for ordinary detectors and light beams, the light scattered > 0.5º will not be detected. Any non-absorbing material must be spatially homogeneous with respect to its dielectric properties in order to become transparent. From an electromagnetic point of view, a defect, or scattering centre, is a spatial region in which a difference in the diagonal dielectric constant (refractive index) is present. The effect of dielectric heterogeneities is manifested through light scattering phenomena, which leads to losses both in the optical quality of the materials and in the total transmitted energy. In ceramics, the main scattering sources are given by the presence of pores, second phase inclusions, rough interfaces… (see Fig. 4.2). Whereas the roughness at interfaces can be minimized by an adequate polishing of the surfaces and the presence of second phases can be also made negligible by selecting pure materials, the presence of pores and the effects of grain boundaries can only be minimized by an adequate processing and sintering. In most cases, and particularly in cubic materials such as yttria, YAG, AlON, spinel, pores are by far the main source of scattering and most of the efforts in the literature have been devoted to minimization of this kind of scattering. In

Transparency represents the ability of materials to allow the transmission of light through them. When traversing a transparent material, a light beam propagates along the same direction before and after traversing the material (see Fig. 4.1a). However, perfectly transparent materials are rare in nature and most of them present the so-called scattering centres. When the concentration of scattering centres is sufficiently large, the material still allows the transmission of light, but the transmitted light beam does not only propagate along the incident direction, but diffuse, off-scattered, beams can be detected; in this case the material becomes translucent (see Fig. 4.1b). In this work, we shall refer only to the transparency of a material, which is measured as the real in-line transmittance. When the amount of scattering centres is higher, the materials scatter so much light that they become

Fig. 4.1. Light propagation through a transparent (a), a translucent (b) and a highly

a b c

Therefore, it becomes necessary to define a measurement technique that allows distinguishing between translucency and transparency. In this case, the relevant parameter is the real in line transmittance (Appetz & Bruggen, 2003) or RIT. In this case, a collimated light beam impinges on the sample and the detector is placed far away, usually 1 m, from the sample. This way, for ordinary detectors and light beams, the light scattered > 0.5º will not be detected. Any non-absorbing material must be spatially homogeneous with respect to its dielectric properties in order to become transparent. From an electromagnetic point of view, a defect, or scattering centre, is a spatial region in which a difference in the diagonal dielectric constant (refractive index) is present. The effect of dielectric heterogeneities is manifested through light scattering phenomena, which leads to losses both in the optical quality of the materials and in the total transmitted energy. In ceramics, the main scattering sources are given by the presence of pores, second phase inclusions, rough interfaces… (see Fig. 4.2). Whereas the roughness at interfaces can be minimized by an adequate polishing of the surfaces and the presence of second phases can be also made negligible by selecting pure materials, the presence of pores and the effects of grain boundaries can only be minimized by an adequate processing and sintering. In most cases, and particularly in cubic materials such as yttria, YAG, AlON, spinel, pores are by far the main source of scattering and most of the efforts in the literature have been devoted to minimization of this kind of scattering. In

**4. Characterization of transparent materials** 

opaque (see Fig. 4.1c).

dispersive, almost opaque (c) material.

non-cubic materials, of which alumina is the most intensively studied material, there is an additional scattering given by the dependence of the value of the refractive index with the orientation of the crystalline grain, which we shall refer to hereafter as scattering due to grains.

Fig. 4.2. Light scattering sources in polycrystalline ceramics.

Whenever a light beam reaches a surface separating two media with different refractive indices (see Figure 4.3), it deviates according to the Snell's law:

$$
\mathbf{n}\_1 \sin \theta\_1 = \mathbf{n}\_2 \sin \theta\_2 \tag{1}
$$

Fig. 4.3. Light refraction at (a) a matrix-pore boundary and (b) between two grains with different crystallographic orientations in a birefringent polycrystalline material.

A model based on geometrical optics was first produced by A. L. Dalisa and R. J. Seymour in which a spread function describing the effect on an incident collimated beam of randomly oriented spatial regions with a difference in the refractive index between zero and ± Δnmax

Sintering to Transparency of Polycrystalline Ceramic Materials 545

and their dimensions verifies the conditions for the second approach (Rayleigh scattering) and, therefore, we'll focus our analysis under that simplification. According to this wellknown approach, the intensity of electromagnetic radiation with wavelength λ scattered by an object of diameter *d* and refractive index *n*, at a distance *R* and angle of scattering

�=��

kept much smaller than the wavelength of the radiation used.

exponentially with thickness, *d*, according to:

2003). The expressions given for those coefficients are:

where rgb (rp) is the radius of the grain (pore); Δn2

��� <sup>=</sup> ������ �� � ����

������� ��� � �� � � � � ���� ����� � � � � � �

�� <sup>=</sup> ��� � �� �� � ���� ����� �

In case a group of N scattering particles is considered, the scattering cross section is given by N times the single scattering cross section. This implies that the scattering due to pores of diameter d in a ceramic is much larger, by a factor of 26, than that due to pores of diameter d/2. If the size of the pores is reduced to half the initial size, the individual scattering is reduced by a factor of 26, whereas the number of scattering centres must be increased by a factor 23 if the total porosity is kept constant. This implies that, in total, by reducing the pore size by a factor of 2, the total dispersion is reduced by a factor 23, almost an order of magnitude. Therefore, in order to improve the real in line transmittance, two approaches can be followed: the porosity is strongly reduced or, as shown above, the size of the pores is

Since the pioneering works by Peelen and Metselaar (Peelen and Metselaar, 1974), lots of efforts have been devoted to analyse and model the effects of the different scattering sources on the real in line transmittance. It is well known that the real in line transmittance decays

where γ, the total scattering, is the sum of the scattering coefficients due to grains and pores: γ = γgb + γp. On the other hand, *Rs* accounts for the total reflected light at the air-material and material-air boundaries. It is well known that the amount of light reflected from a surface between two media (Born & Wolf, 2005) with refractive indices n1 and n2 at normal incidence is given by Rs=(n1-n2)2/(n1+n2)2. In this case, one of the media is air (n1=1) and the other medium is the material under study (n2=n). R. Appetz and M. P. B. van Bruggen calculated the light scattering coefficients due to pores and different grain orientations in non-cubic materials, considering just one pore size and one grain size (Appetz & Bruggen,

� and �� <sup>=</sup> �

indices between grains in birefringent materials (pore and matrix); p is the total porosity of the material and λ0 is the wavelength of light used in vacuum. More recently, this analysis was extended to a distribution of pore and particle sizes (Suarez et al., 2009). In Figure 4.5,

� � ��� � ����� �� � ���

gb (Δn2

��� = (����)� � ����(���) (4)

The Rayleigh scattering cross section for a single particle can be expressed as:

given by:

θis

(2)

� (5)

p) is the difference in the refractive

(3)

was developed for ferroelectric ceramics (Dalisa & Seymour, 1973). This model was further applied by R. Appetz et al. to study the off-specular transmittance in polycrystalline alumina (Appetz & Bruggen, 2003), a birefringent material for which n1 can be taken as 1.768 and n2 as 1.76. Considering a grain boundary placed at 45º, the deviation of the light beam at this interface is calculated to be around 0.26º. If a beam light traverses a 1 mm thick polycrystalline alumina sample, in which the grain size is around 0.5 μm, the number of refractions that a light beam undergoes is not below several thousands. This gives an idea of the large scattering undergone by the beam. This discussion also illustrates why the larger the number of pores (for the same pore size), the smaller the real in line transmittance. However, following the same reasoning, one would expect the real in line transmittance to be higher for increasing grain size, whereas the experimental observations (Appetz & Bruggen, 2003) indicate that the amount of light scattered out of the normal incidence is larger in a sample with a grain size of 20 μm than in one with a grain size of 1 μm (Figure 4.4). This indicates that the geometrical model used above is not valid anymore to study the scattering of objects with dimensions of the order of the wavelength of the light.

Fig. 4.4. Angular scattering profiles corresponding to 0.8 mm thick polycrystalline alumina samples with 1 μm (solid line) and 20 μm (dotted line) grain size (taken from Appetz & Bruggen, 2003).

In systems with a few defects and by using the Mie equation (Mie, 1908), it is possible to determine the intensity and directionality of the scattered radiation. This formulation is derived from the Maxwell equations and it is, therefore, valid for arbitrary defect sizes. However, as the Mie equation is complex to handle, it becomes helpful to use approximations. Two approximations are often used (Bohren & Huffmann, 2010; van de Hulst, 1957): Rayleigh-Gans-Debye scattering for large particles: 2π*r*>>λ and Rayleigh scattering for small particles: 2π*r*<<λ, with *r* being the radius of the scattering centre. When dealing with transparent materials or materials close to transparency, the number of pores

was developed for ferroelectric ceramics (Dalisa & Seymour, 1973). This model was further applied by R. Appetz et al. to study the off-specular transmittance in polycrystalline alumina (Appetz & Bruggen, 2003), a birefringent material for which n1 can be taken as 1.768 and n2 as 1.76. Considering a grain boundary placed at 45º, the deviation of the light beam at this interface is calculated to be around 0.26º. If a beam light traverses a 1 mm thick polycrystalline alumina sample, in which the grain size is around 0.5 μm, the number of refractions that a light beam undergoes is not below several thousands. This gives an idea of the large scattering undergone by the beam. This discussion also illustrates why the larger the number of pores (for the same pore size), the smaller the real in line transmittance. However, following the same reasoning, one would expect the real in line transmittance to be higher for increasing grain size, whereas the experimental observations (Appetz & Bruggen, 2003) indicate that the amount of light scattered out of the normal incidence is larger in a sample with a grain size of 20 μm than in one with a grain size of 1 μm (Figure 4.4). This indicates that the geometrical model used above is not valid anymore to study the

scattering of objects with dimensions of the order of the wavelength of the light.


Scattering angle (degrees)

Fig. 4.4. Angular scattering profiles corresponding to 0.8 mm thick polycrystalline alumina samples with 1 μm (solid line) and 20 μm (dotted line) grain size (taken from Appetz &

In systems with a few defects and by using the Mie equation (Mie, 1908), it is possible to determine the intensity and directionality of the scattered radiation. This formulation is derived from the Maxwell equations and it is, therefore, valid for arbitrary defect sizes. However, as the Mie equation is complex to handle, it becomes helpful to use approximations. Two approximations are often used (Bohren & Huffmann, 2010; van de Hulst, 1957): Rayleigh-Gans-Debye scattering for large particles: 2π*r*>>λ and Rayleigh scattering for small particles: 2π*r*<<λ, with *r* being the radius of the scattering centre. When dealing with transparent materials or materials close to transparency, the number of pores

 Grain size: 1 μm Grain size: 20 μm

10-5

Bruggen, 2003).

10-4

10-3

Transmittance

10-2

10-1

100

and their dimensions verifies the conditions for the second approach (Rayleigh scattering) and, therefore, we'll focus our analysis under that simplification. According to this wellknown approach, the intensity of electromagnetic radiation with wavelength λ scattered by an object of diameter *d* and refractive index *n*, at a distance *R* and angle of scattering θ is given by:

$$I = I\_0 \frac{1 + \cos^2 \theta}{2\mathcal{R}^2} \left(\frac{2\pi}{\lambda}\right)^4 \left(\frac{n^2 - 1}{n^2 + 2}\right)^2 \left(\frac{d}{2}\right)^6 \tag{2}$$

The Rayleigh scattering cross section for a single particle can be expressed as:

$$
\sigma\_s = \frac{2\pi^5}{3} \frac{d^6}{\lambda^4} \left( \frac{n^2 - 1}{n^2 + 2} \right)^2 \tag{3}
$$

In case a group of N scattering particles is considered, the scattering cross section is given by N times the single scattering cross section. This implies that the scattering due to pores of diameter d in a ceramic is much larger, by a factor of 26, than that due to pores of diameter d/2. If the size of the pores is reduced to half the initial size, the individual scattering is reduced by a factor of 26, whereas the number of scattering centres must be increased by a factor 23 if the total porosity is kept constant. This implies that, in total, by reducing the pore size by a factor of 2, the total dispersion is reduced by a factor 23, almost an order of magnitude. Therefore, in order to improve the real in line transmittance, two approaches can be followed: the porosity is strongly reduced or, as shown above, the size of the pores is kept much smaller than the wavelength of the radiation used.

Since the pioneering works by Peelen and Metselaar (Peelen and Metselaar, 1974), lots of efforts have been devoted to analyse and model the effects of the different scattering sources on the real in line transmittance. It is well known that the real in line transmittance decays exponentially with thickness, *d*, according to:

$$RIT = (1 - R\_s)^2 \cdot \exp(-\gamma d) \tag{4}$$

where γ, the total scattering, is the sum of the scattering coefficients due to grains and pores: γ = γgb + γp. On the other hand, *Rs* accounts for the total reflected light at the air-material and material-air boundaries. It is well known that the amount of light reflected from a surface between two media (Born & Wolf, 2005) with refractive indices n1 and n2 at normal incidence is given by Rs=(n1-n2)2/(n1+n2)2. In this case, one of the media is air (n1=1) and the other medium is the material under study (n2=n). R. Appetz and M. P. B. van Bruggen calculated the light scattering coefficients due to pores and different grain orientations in non-cubic materials, considering just one pore size and one grain size (Appetz & Bruggen, 2003). The expressions given for those coefficients are:

$$\gamma\_{gb} = \frac{3r\_{gb}\pi^2}{\lambda\_0^2} \Delta n\_{gb}^2 \text{ and } \gamma\_p = \frac{p}{\frac{4}{3}\pi r\_p^3} \frac{3r\_p\pi^2}{\lambda\_0^2} \Delta n\_p^2 \tag{5}$$

where rgb (rp) is the radius of the grain (pore); Δn2 gb (Δn2 p) is the difference in the refractive indices between grains in birefringent materials (pore and matrix); p is the total porosity of the material and λ0 is the wavelength of light used in vacuum. More recently, this analysis was extended to a distribution of pore and particle sizes (Suarez et al., 2009). In Figure 4.5,

Sintering to Transparency of Polycrystalline Ceramic Materials 547

The simulations shown above indicate the critical role of pores when transparent ceramics are being pursued. However, the situation can become more complex when the crystalline structure of the material considered is not cubic. In this case, the diagonal elements of the dielectric tensor will be, in general, different. In other words, the refractive index will depend on the crystalline orientation of each individual grain. For example, due to symmetry reasons, in alumina, two of the diagonal elements of the dielectric tensor are equal, but the third one is different (εxx=εyy≠εzz). In alumina, this leads to a difference in the refractive index of only 0.008, but enough to induce a large scattering, particularly in the visible range. As shown in Figure 4.7a, the larger the grain size, the larger the scattering and, therefore, the smaller the RIT. Also, analogously to the porosity, the scattering becomes more critical at shorter wavelengths. It is shown in Figure 4.7a that whereas the RIT at 400 nm for a polycrystalline material does not reach 10%, it is close to 40% at 700 nm. Therefore, the grain size will be very important when materials are developed for the visible range and will not be so critical in devices operating in the infrared range. Following the same formalism, it can be shown that the RIT only falls to 60% at 1.5 microns and 70% at 2 microns when the average grain size is 2 microns in polycrystalline alumina. All RIT values shown so far have been calculated considering texture less materials. However, different routes have shown the possibility to induce some texture in the polycrystalline materials (Salem et al., 1989; Mao et al., 2008; Uchikoshi et al., 2004). Figure 4.7b shows how at two wavelengths, 400 and 700 nm, the RIT increases with increasing texture. In this simulation, texture = 0 corresponds to a perfectly polycrystalline material whereas texture = 1 corresponds to a fully textured, analogous to a single crystal, material. It is worth noting that no porosity has been considered in the simulations and that even so, at 400 nm, the RIT strongly decays from its maximum, around 80%, to only 20% due to a random orientation of the crystallites forming the material. The behaviour at 700 nm is similar, but the decay is not so abrupt, indicating again the strong wavelength dependence of the phenomenon. For these simulations, a grain size of only 500 nm has been considered, which is not easy to obtain in conventionally sintered alumina materials. Usual grain sizes are of the order of several μm, which leads to even more drastic reductions in the RIT, as can be deduced from

0

20

40

RIT (%)

60

80

(b)

100

0 10 20 30 40 50 60 70 80 90 100

Porosity: 0.0005 Density: 99.95% Lambda: 590 nm

Pore size (nm)

0.0 0.1 0.2 0.3 0.4 0.5

pore radius: 10 nm Lambda: 590 nm

Fig. 4.6. Simulations considering different porosities and pore sizes.

Porosity (%)

0

Figure 4.7a.

20

40

RIT (%)

60

80

(a)

100

the effect of considering a distribution of pores instead of a single pore is given. For this simulation, a Poisson pore size distribution with an average radius of 30 nm was taken into account. It can be observed that the main differences take place at short wavelengths, that is, in the visible part of the spectrum for the numbers considered in this example.

Fig. 4.5. Transmittance spectra in a material with a single pore size or a Poisson distribution of pore sizes.

In Figure 4.6, the effect of changing both the porosity for a fixed pore size (4.6.a) and the pore size at a fixed porosity (4.6.b) is simulated according to the formalism developed by Suarez et al. (Suarez et al., 2009). Some remarkable features should be highlighted: as shown in Figure 4.6.a, for a pore radius of 10 nm, in a 1 mm thick material and at a wavelength of 600 nm, i. e., in the visible range, a porosity of 0.5%, which corresponds to a density of the material of 99.5%, implies that the RIT decays to almost zero. Actually, values of the porosity over 0.3% lead to RIT values so small that their detection becomes very complicated for usual detectors. On the other hand, at the same wavelength and considering a porosity of only 0.05%, which corresponds to a density of 99.95%, it is shown that pore sizes over, approximately, 50 nm lead to very small RIT values. These two results combined imply that achieving a high density (>99.9%) is not usually enough to obtain a transparent material. It is also necessary to keep the values of pores below a given value that, according to these simulations, can be established below 30 nm.

the effect of considering a distribution of pores instead of a single pore is given. For this simulation, a Poisson pore size distribution with an average radius of 30 nm was taken into account. It can be observed that the main differences take place at short wavelengths, that is,

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

 Single pore size Pore size distribution

Average pore radius: 30 nm Refractive index: 1.76

Porosity: 0.04% Density: 99.96%

Wavelength (μm)

Fig. 4.5. Transmittance spectra in a material with a single pore size or a Poisson distribution

In Figure 4.6, the effect of changing both the porosity for a fixed pore size (4.6.a) and the pore size at a fixed porosity (4.6.b) is simulated according to the formalism developed by Suarez et al. (Suarez et al., 2009). Some remarkable features should be highlighted: as shown in Figure 4.6.a, for a pore radius of 10 nm, in a 1 mm thick material and at a wavelength of 600 nm, i. e., in the visible range, a porosity of 0.5%, which corresponds to a density of the material of 99.5%, implies that the RIT decays to almost zero. Actually, values of the porosity over 0.3% lead to RIT values so small that their detection becomes very complicated for usual detectors. On the other hand, at the same wavelength and considering a porosity of only 0.05%, which corresponds to a density of 99.95%, it is shown that pore sizes over, approximately, 50 nm lead to very small RIT values. These two results combined imply that achieving a high density (>99.9%) is not usually enough to obtain a transparent material. It is also necessary to keep the values of pores below a given value that, according

0

to these simulations, can be established below 30 nm.

of pore sizes.

10

20

30

40

Transmittance (%)

50

60

70

80

in the visible part of the spectrum for the numbers considered in this example.

Fig. 4.6. Simulations considering different porosities and pore sizes.

The simulations shown above indicate the critical role of pores when transparent ceramics are being pursued. However, the situation can become more complex when the crystalline structure of the material considered is not cubic. In this case, the diagonal elements of the dielectric tensor will be, in general, different. In other words, the refractive index will depend on the crystalline orientation of each individual grain. For example, due to symmetry reasons, in alumina, two of the diagonal elements of the dielectric tensor are equal, but the third one is different (εxx=εyy≠εzz). In alumina, this leads to a difference in the refractive index of only 0.008, but enough to induce a large scattering, particularly in the visible range. As shown in Figure 4.7a, the larger the grain size, the larger the scattering and, therefore, the smaller the RIT. Also, analogously to the porosity, the scattering becomes more critical at shorter wavelengths. It is shown in Figure 4.7a that whereas the RIT at 400 nm for a polycrystalline material does not reach 10%, it is close to 40% at 700 nm. Therefore, the grain size will be very important when materials are developed for the visible range and will not be so critical in devices operating in the infrared range. Following the same formalism, it can be shown that the RIT only falls to 60% at 1.5 microns and 70% at 2 microns when the average grain size is 2 microns in polycrystalline alumina. All RIT values shown so far have been calculated considering texture less materials. However, different routes have shown the possibility to induce some texture in the polycrystalline materials (Salem et al., 1989; Mao et al., 2008; Uchikoshi et al., 2004). Figure 4.7b shows how at two wavelengths, 400 and 700 nm, the RIT increases with increasing texture. In this simulation, texture = 0 corresponds to a perfectly polycrystalline material whereas texture = 1 corresponds to a fully textured, analogous to a single crystal, material. It is worth noting that no porosity has been considered in the simulations and that even so, at 400 nm, the RIT strongly decays from its maximum, around 80%, to only 20% due to a random orientation of the crystallites forming the material. The behaviour at 700 nm is similar, but the decay is not so abrupt, indicating again the strong wavelength dependence of the phenomenon. For these simulations, a grain size of only 500 nm has been considered, which is not easy to obtain in conventionally sintered alumina materials. Usual grain sizes are of the order of several μm, which leads to even more drastic reductions in the RIT, as can be deduced from Figure 4.7a.

Sintering to Transparency of Polycrystalline Ceramic Materials 549

measurements to a given thickness. The expression that relates the transmittance at a normalized thickness d1 considering a measurement performed on a material with a

�� � ������]

In conclusion, the interest on transparent ceramic materials relies on the extraordinary combination of high mechanical performance, chemical resistance/inertness… and little or none absorption in different ranges of the electromagnetic spectrum from infrared to near ultra violet. It has been shown that obtaining a transparent ceramic material implies sintering to a density, very close to the theoretical density (≥99.9%), and keeping the few pores left with a pore size below 30 nm. In order to achieve this aim, attention must be paid not only to the raw materials: adequate granulometry, particle size… but it becomes also critical to control the packing of the powders in the green body. Usually, large defects present in the green bodies cannot be removed during the sintering process, and therefore, forming step is decisive for the preparation of transparent materials. Moreover, it has been shown how an adequate choice of the sintering technique combined with a tailoring of the starting powder is critical when an accurate control of the microstructural features is required. Finally, different analysis techniques of the transmittance spectra have been given from which average structural parameters such as pore size, porosity and grain size and

Álvarez-Clemares I., Mata-Osoro G., Fernández A., López-Esteban S., Pecharromán C.,

Amato I., Baudrocco F. & Martorana D. (1976). Evaluation of freeze drying and spray drying

Anderson R.C. (1998). Transparent Yttria-based ceramics and method for producing same.

Appetz R. & Van Bruggen M.P.B. (2003). Transparent alumina: a light scattering model.

Bennison S.J. & Harmer M.P. (1990a). A history of the role of MgO in the sintering of alfaalumina. Ceramic Transactions, American Ceramic Society, 7, 13-49. Bennison S.J. & Harmer M.P. (1990b). Effect of magnesia on surface diffusion in sapphire

Bohren C. F. & Huffmann D.R. (2010). Absorption and scattering of light by small particles,

Journal of the American Ceramic Society, 86, 3, 480-486.

Wiley-Interscience, ISBN 3527406646, New York

Palomares J., Torrecillas R. & Moya J.S. (2010). Transparent alumina/ceria nanocomposites by spark plasma sintering. Advanced Engineering Materials, 12,

processes for preparing transparent alumina. Material Science and Engineering, 26,

and the role of magnesia in the sintering of alumina. Journal of the American

�� �� � � � �(��)

(����)��

�� �� �

(6)

��

thickness d2 is given by

**5. Conclusions** 

**6. References** 

11, 1154-1160

U. S. Patent 3545987.

Ceramic Society, 73, 4, 833-837

1, 73-78.

�(��)

texture in birefringent ceramics can be extracted.

(����)� � ����� � �����·

Fig. 4.7. Simulations for birefringent materials with different textures and grain sizes.

C. Pecharroman et al. developed a theoretical model (Pecharroman et al., 2009) for light scattering due to polycrystalline aggregates of uniaxial spheres within the Rayleigh-Gans-Debye approximation. This model shows that the scattering efficiency of each individual grain depends linearly on the grain size and on the texture of the samples. As indicated above, the most critical scattering is that due to the pores and this explains why common technical alumina ceramics which are considered dense at relative density over 99.5 % are opaque (white) when sintered in air, even if high purity raw materials are used. However, provided that the scattering due to pores decreases as (d/λ)4, in order to make a material transparent it is not necessary to completely remove all the scattering centres, which is a cumbersome task, but it is enough to keep them all below a certain critical size. As an approximate rule, it is considered that the defects should be smaller than λ/10. In particular, for the visible range, in which the considered wavelengths range from 400 to 750 nm, the pore size should not be over a few tens of nm. Even so, the total porosity should never be over 0.05%. For this reason care must be taken not only during sintering, but also during processing and preparation of the green bodies. A uniform packing of the powders in green bodies has been shown in the literature to be critical to obtain low porosity ceramics. This is of particular importance when working with nanoceramics, as the attractive Van der Waals forces between particles are responsible for the high tendency of the nanoparticles to agglomerate. The presence of agglomerates leads to local differences in density which changes the sintering behaviour of the body leading to microcracks and microstructural effects such as residual pores and local coarsening and, therefore, to a large scattering. On the one hand, the raw powder should be very fine-grained in order to increase the sintering activity resulting in the elimination of residual porosity. Also, the structural homogeneity of sintering bodies is most important for a minimum of flaws that are detrimental to both optical and mechanical properties. This request for homogeneity makes the use of nanopowders < 100 nm still difficult. Casting methods are particularly suited to process raw powders with particle sizes between 100 and 150 nm.

Finally, it is obvious that preparing materials with the same thickness for a set of measurements is extremely difficult. For this reason, it is important to normalize measurements to a given thickness. The expression that relates the transmittance at a normalized thickness d1 considering a measurement performed on a material with a thickness d2 is given by

$$\frac{\Gamma(d\_1)}{(1-R\_3)^2} = e^{-\gamma d\_1} = e^{-\gamma d\_1 \frac{d\_2}{d\_2}} = \left[e^{-\gamma d\_2}\right]^{d\_1} \Big/\_{d\_2} = \left[\frac{\Gamma(d\_2)}{(1-R\_3)^2}\right]^{d\_1} \Big/\_{d\_2} \tag{6}$$

### **5. Conclusions**

548 Sintering of Ceramics – New Emerging Techniques

C. Pecharroman et al. developed a theoretical model (Pecharroman et al., 2009) for light scattering due to polycrystalline aggregates of uniaxial spheres within the Rayleigh-Gans-Debye approximation. This model shows that the scattering efficiency of each individual grain depends linearly on the grain size and on the texture of the samples. As indicated above, the most critical scattering is that due to the pores and this explains why common technical alumina ceramics which are considered dense at relative density over 99.5 % are opaque (white) when sintered in air, even if high purity raw materials are used. However, provided that the scattering due to pores decreases as (d/λ)4, in order to make a material transparent it is not necessary to completely remove all the scattering centres, which is a cumbersome task, but it is enough to keep them all below a certain critical size. As an

for the visible range, in which the considered wavelengths range from 400 to 750 nm, the pore size should not be over a few tens of nm. Even so, the total porosity should never be over 0.05%. For this reason care must be taken not only during sintering, but also during processing and preparation of the green bodies. A uniform packing of the powders in green bodies has been shown in the literature to be critical to obtain low porosity ceramics. This is of particular importance when working with nanoceramics, as the attractive Van der Waals forces between particles are responsible for the high tendency of the nanoparticles to agglomerate. The presence of agglomerates leads to local differences in density which changes the sintering behaviour of the body leading to microcracks and microstructural effects such as residual pores and local coarsening and, therefore, to a large scattering. On the one hand, the raw powder should be very fine-grained in order to increase the sintering activity resulting in the elimination of residual porosity. Also, the structural homogeneity of sintering bodies is most important for a minimum of flaws that are detrimental to both optical and mechanical properties. This request for homogeneity makes the use of nanopowders < 100 nm still difficult. Casting methods are particularly suited to process raw

Finally, it is obvious that preparing materials with the same thickness for a set of measurements is extremely difficult. For this reason, it is important to normalize

Fig. 4.7. Simulations for birefringent materials with different textures and grain sizes.

approximate rule, it is considered that the defects should be smaller than

0

20

40

Transmittance (%)

60

80

(b)

100

0.0 0.2 0.4 0.6 0.8 1.0

Texture

λ

/10. In particular,

 λ: 400 nm λ: 700 nm

λ: 400 nm λ: 700 nm

0 500 1000 1500 2000

Grain size (nm)

powders with particle sizes between 100 and 150 nm.

0

20

40

Transmittance (%)

60

80

(a)

100

In conclusion, the interest on transparent ceramic materials relies on the extraordinary combination of high mechanical performance, chemical resistance/inertness… and little or none absorption in different ranges of the electromagnetic spectrum from infrared to near ultra violet. It has been shown that obtaining a transparent ceramic material implies sintering to a density, very close to the theoretical density (≥99.9%), and keeping the few pores left with a pore size below 30 nm. In order to achieve this aim, attention must be paid not only to the raw materials: adequate granulometry, particle size… but it becomes also critical to control the packing of the powders in the green body. Usually, large defects present in the green bodies cannot be removed during the sintering process, and therefore, forming step is decisive for the preparation of transparent materials. Moreover, it has been shown how an adequate choice of the sintering technique combined with a tailoring of the starting powder is critical when an accurate control of the microstructural features is required. Finally, different analysis techniques of the transmittance spectra have been given from which average structural parameters such as pore size, porosity and grain size and texture in birefringent ceramics can be extracted.

#### **6. References**


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**24** 

*1Tunisie 2,3France* 

**Effects of the Microstructure** 

*1Sfax University, LaMaCoP, BP 1171, Sfax 3000,* 

**Properties of Alumina** 

Zarbout Kamel1, Moya Gérard2,

**Induced by Sintering on the Dielectric** 

Si Ahmed Abderrahmane2, Damamme Gilles3 and Kallel Ali1

*3Commissariat à l'Energie Atomique, DAM Ile-de-France, Bruyère-le-Châtel* 

The dielectric properties of undoped corundum, α-alumina, of different kinds (single crystal or polycrystalline obtained by solid state sintering) have been the subject of numerous studies concerning, in particular, the breakdown strength and the charging behavior (Haddour et al., 2009; Liebault et al., 2001; Si Ahmed et al., 2005; Suharyanto et al., 2006; Thome et al., 2004; Touzin et al., 2010; Zarbout et al., 2008, 2010). The common feature pointed out by most of these investigations is the conspicuous role played by the microstructure and the impurities. It is also established that the microstructure induced by the sintering process (grain size and porosity) goes concomitantly along with impurities segregation at grain boundaries and/or the development of defects in the lattice (Chiang et al., 1996; Gavrilov et al., 1999; Lagerlöf & Grimes, 1998; Moya et al., 2003). To some extent, for a given composition, these evolutions can be governed by the sintering conditions, for

The breakdown strength is a key parameter for the reliability of dielectrics and in particular of microelectronic insulator components. This parameter is intimately linked to the charging properties as breakdown originates from the enhancement of the density of trapped charges, which stems from the competition between charge trapping and conduction (Blaise & Le Gressus, 1991; Le Gressus et al., 1991; Liebault et al., 2001; Haddour et al., 2009). The charges can be either generated by irradiation or injected through interfaces via an applied voltage. Charge trapping can occur around intrinsic point defects, defects induced by the dissolution of impurities, defects associated with grain boundaries interfaces and dislocations (Kolk & Heasell, 1980). We must also keep in mind that trapping in insulators gives rise to polarization and lattice deformation allowing energy accumulation within the material (Blaise & Le Gressus, 1991; Le Gressus et al., 1991; Stoneham, 1997). As a result, if some critical density of trapped charges (or some critical electric field) is reached, external stresses (thermal, electrical or mechanical) can trigger a collective relaxation process corresponding

instance the firing cycle in the case of solid state sintering (Chiang et al., 1996).

**1. Introduction** 

*2Aix-Marseille University, Im2np, UMR-CNRS 6242, Marseille,* 

