*3.1.3. Sintering mechanisms*

source, the sintering mechanisms are also a contributing factor induced by driving forces. Figure 6 shows the possible forces involved in the sintering process: surface free energy,

**Figure 6.** The three main drivers for solid densification: surface free energy, applied pressure, and chemical reaction [38].

**Figure 7.** Diagram flow of vacancies on the surface. The atoms flow is opposite to the vacancy [38].

The variation of free energy during sintering is represented by Eq. 11:

 dg

d

in Figure 7.

The surface energy is related to the surfaces curve and characterized by vacancies and gaps. The surfaces energy is the main force that sinters the material by mass flow through the region of higher concentration to a lower concentration region where vacancies and gaps, as shown

*G dA dA SS SS SV SV*

 dg

= + ò ò (11)

applied external pressure, and chemical reaction [38].

34 Advanced Ceramic Processing

In polycrystalline materials, the mass transport ways that are responsible for sintering are diffusion via crystal lattice, surface diffusion, volume diffusion, plastic flow, and evaporation– condensation. Figure 8 shows all mass transport paths arrive at the point of contact between two particles [38].

**Figure 8.** Mass transport mechanism solid and viscous sintering [38].

In Figure 8, the first three mechanisms do not lead to an alignment of the mass centers of the particles and therefore are non-densifying mechanisms. Thus, the mechanisms that start on the volume of material to the neck that increase in the neck and decrease the distance between the particles are densifying mechanisms [39].

#### *3.1.4. Stages of sintering*

The sintering mechanisms occur by three successive or simultaneously stages divided into initial, intermediate, and final stages. In some cases, there is the zero stage, which corresponds to particle rearrangement stage for subsequent joining by spot contact called necks [40]. The initial stage consists of particles rounding, formation of necks with low grain growth, and significant reduction in surface area and porosity. This stage progresses until the point where the necks interfere with each other. This stage corresponds to the point where the dihedral angle of equilibrium is reached. For the system with the green density of ~60%, this corresponds to a linear shrinkage of 3% to 5% [36]. It is possible to develop a general equation of the sintering kinetics for the initial stage. The geometric model for the development of this mathematical relationship is illustrated in Figure 9:

**Figure 9.** Frenkel's model for early-stage sintering viscous flow [41].

The two spheres of the Frenkel's model use the concept of viscous flow of atoms that relates the vacancy diffusion coefficient *D*v, the volume of the atom or vacancy *Ω*, and vacancy concentration gradient per unit area of the material (d*C*v/d*x*), as shown in the following equation [41]:

$$J\_a = \frac{D\_v}{\Omega} \frac{d\mathbf{C}\_v}{d\mathbf{x}} \tag{12}$$

Thus, the transported mass volume as a function of time can be given by [41]

New Approaches to Preparation of SnO2-Based Varistors — Chemical Synthesis, Dopants, and Microwave Sintering http://dx.doi.org/10.5772/61206 37

$$\frac{dV}{dt} = J\_a A\_{\text{gb}} \Omega \tag{13}$$

were *Agb* =2*πX δSV* is equal to the cross-sectional area where diffusion occurs, and *X* is the radius of the neck.

In Figure 8, the first three mechanisms do not lead to an alignment of the mass centers of the particles and therefore are non-densifying mechanisms. Thus, the mechanisms that start on the volume of material to the neck that increase in the neck and decrease the distance between

The sintering mechanisms occur by three successive or simultaneously stages divided into initial, intermediate, and final stages. In some cases, there is the zero stage, which corresponds to particle rearrangement stage for subsequent joining by spot contact called necks [40]. The initial stage consists of particles rounding, formation of necks with low grain growth, and significant reduction in surface area and porosity. This stage progresses until the point where the necks interfere with each other. This stage corresponds to the point where the dihedral angle of equilibrium is reached. For the system with the green density of ~60%, this corresponds to a linear shrinkage of 3% to 5% [36]. It is possible to develop a general equation of the sintering kinetics for the initial stage. The geometric model for the development of this mathematical

The two spheres of the Frenkel's model use the concept of viscous flow of atoms that relates the vacancy diffusion coefficient *D*v, the volume of the atom or vacancy *Ω*, and vacancy concentration gradient per unit area of the material (d*C*v/d*x*), as shown in the following

*v v*

*D dC <sup>J</sup> dx* <sup>=</sup> <sup>W</sup> (12)

*a*

Thus, the transported mass volume as a function of time can be given by [41]

the particles are densifying mechanisms [39].

relationship is illustrated in Figure 9:

**Figure 9.** Frenkel's model for early-stage sintering viscous flow [41].

equation [41]:

*3.1.4. Stages of sintering*

36 Advanced Ceramic Processing

Assuming that the decrease in surface energy of the system is equivalent to the energy dissipated through the material flow, then it is possible to derive several equations relating the radius of the neck and ball as a function of sintering time [38,42]:

$$\left(\frac{X}{a}\right)^{\text{w}} = \frac{H}{a^{\text{u}}}t\tag{14}$$

where *m* and *n* are the sintering mechanisms, *H* is a function that varies with parameters such as diffusion rate, surface tension, atom or vacancy size, and *a* is the radius of the sphere.

Many aspects can be studied from the kinetic equations, as densification rate, determination of sintering mechanisms, and activation energy. The equation developed by Coble allows to estimate the sintering mechanisms for the initial stage, based on the two spheres Frenkel's model, as indicated in the Eq. 15 [43]:

$$Y^n = k\_0 \exp\left(\frac{-Q}{RT}\right)t\tag{15}$$

where *n* = 1, 2, 3, or 4, indicating the predominant mechanism of viscous flow, surface diffusion, and diffusion via grain boundary diffusion and via crystal lattice, and *Y* is the linear shrinkage of the sample, *Q* is the activation energy, *R* is the gas constant real, *T* is the temperature, and *t* is time.

The intermediate stage initiates densifying mechanisms as volumetric diffusion by crystal lattice in which there is rapid grain growth, shrinkage pore and increased in the density of the material up to ~90% of the theoretical density. Whereas there is grain growth, the model for the initial stage does not fit this stage. The final stage is characterized by the elimination of residual pores with little or no densification, but grain growth is observed. For the determi‐ nation of sintering mechanisms, intermediate and final stages are used in the model-based grain growth [44]:

$$\mathbf{G}'' - \mathbf{G}''\_0 = k\_0 \exp\left(\frac{-Ea\_b}{RT}\right)\mathbf{t} \tag{16}$$

where *G* is the average grain size, *Eab* is the activation energy for moving contour or grain growth, *n* is the sintering mechanism when valley 3 is spread via reticulum and 4 is broadcast via grain boundary, and *k*0 is a constant that depends on temperature and sintering mecha‐ nisms [41,43,44].

### *3.1.5. Sintering model for thick films*

Most of the kinetic studies of SnO2-based ceramic are developed to oxide mixed synthesis compressed into pellets, where significant amounts of mass are used. However, the appearance of thick and thin films makes possible the integration of smaller electric devices, and thus new techniques for the synthesis and deposition of powders on conductive and insulating rigid substrates have been studied.

The sintering of films has been increasingly used for applications in sensors, fuel cells, or photo catalysis that requires porous films [45,46]. This application is based on the fact that sintering occurs on rigid substrates such as viscous flow, wherein the voltage-limiting densification of the material is the force of attraction between the substrate and the deposited material particles [47,48]. The model used for understanding the sintering of thin films is based on Scherer and Garino's studies where the rate of densification of the film is delayed by the substrate, as in Eq. 17 [38,41]:

$$
\left(\frac{\dot{\rho}}{\rho}\right)\_c = -\left[\frac{1+v\_p}{3\left(1-v\_p\right)}\right] 3\dot{x}\_f \tag{17}
$$

The sintering mechanisms remain the same; however, the densification rate is retarded by tension caused by the substrate, like as the system would be sintered followed viscous sintering mechanism, as with glass.
