**4. Influence of triple junctions on grain boundary motion**

According Czubayco et al., during the formation of granular structure of a polycrystalline material, both grain boundaries and triple junctions influence in characteristics. In the past, just the of grain boundaries motion were studied, while the influence of triple junctions on grain boundary has not attention necessary. In the last years, the velocity of the junction motion, the shape of the intersecting grain boundaries must be measured. Moreover, the steady-state motion of a grain boundary system with a triple junction is only possible in a very narrow range of geometrical boundary configurations. It is usually assumed that triple junctions do not influence the motion of the adjoining grain boundaries and that their role is reduced to control the thermodynamic equilibrium angles at the junction during the boundary motion. A specific mobility of triple junctions was first introduced by Shvindlerman and co-workers (Galina et al., 1987), who considered the steady-state motion of a grain boundary system with a triple junction. The geometry of the used boundary system is shown in Fig. 2. The boundaries of this system are perpendicular to the plane of the diagram, and far from the triple junction they run parallel to one another and to the x-axis.

Fig. 2. Geometry of the grain boundary system with triple junction in the course of steady state motion (From: Czubayko et al., 1998).

The three boundaries of the system are considered identical, in particular their surface tension *σ* and their mobility *m*GB. Furthermore, it is assumed that *m*GB and *σ* are independent of the inclination of the grain boundaries. These assumptions define the problem to be symmetric with regard to the x-axis. With these simplifications some very important

difficulties in maintaining sources and sinks to accommodate point defects 21±23. This leads to a threshold energy or stress, of the order of 2g/G. For a grain size of 100 nm, this amount to 20MPa, which is rather substantial compared to capillary pressure and could be the cause for the suppression. This effect should diminish at larger grain sizes, allowing the kinetic window to extend to lower temperatures. Therefore, in exploiting the difference in the kinetics of grain-boundary diffusion and grain-boundary migration to achieve densification without growth at lower temperatures, it is still advisable to utilize dopants to `tune' the

According Czubayco et al., during the formation of granular structure of a polycrystalline material, both grain boundaries and triple junctions influence in characteristics. In the past, just the of grain boundaries motion were studied, while the influence of triple junctions on grain boundary has not attention necessary. In the last years, the velocity of the junction motion, the shape of the intersecting grain boundaries must be measured. Moreover, the steady-state motion of a grain boundary system with a triple junction is only possible in a very narrow range of geometrical boundary configurations. It is usually assumed that triple junctions do not influence the motion of the adjoining grain boundaries and that their role is reduced to control the thermodynamic equilibrium angles at the junction during the boundary motion. A specific mobility of triple junctions was first introduced by Shvindlerman and co-workers (Galina et al., 1987), who considered the steady-state motion of a grain boundary system with a triple junction. The geometry of the used boundary system is shown in Fig. 2. The boundaries of this system are perpendicular to the plane of the diagram, and far from the

Fig. 2. Geometry of the grain boundary system with triple junction in the course of steady

The three boundaries of the system are considered identical, in particular their surface tension *σ* and their mobility *m*GB. Furthermore, it is assumed that *m*GB and *σ* are independent of the inclination of the grain boundaries. These assumptions define the problem to be symmetric with regard to the x-axis. With these simplifications some very important

**4. Influence of triple junctions on grain boundary motion** 

triple junction they run parallel to one another and to the x-axis.

state motion (From: Czubayko et al., 1998).

overall kinetics.

features of the motion of this system can be established. (a) A steady-state motion of the whole system is possible indeed. (b) The dimensionless criterion Λ:

$$
\Lambda = \frac{m^{\text{T}}a}{m^{\text{GB}}} = \frac{2\theta}{2\cos\theta - 1} \tag{3}
$$

describes the drag influence of the triple junction on the motion of the entire boundary system. For Λ >>1 the junction does not drag the motion of the boundary system, and the angle θ tends to the equilibrium value *π*/3. In such a case the velocity *ν* of the motion of the entire boundary system is independent of the mobility of the triple junction and is determined by the grain boundary mobility and the acting driving force:

$$\nu = \frac{2\pi m^{\text{GB}}\sigma}{3a} \tag{4}$$

In contrast, for Λ<<1 the motion of the system is controlled by the motion of the triple junction and the angle θ tends to zero. The velocity depends only on the triple junction mobility and the grain boundary surface tension *σ*:

$$\mathcal{V} = \sigma m^{T} \tag{5}$$

Owing to the fact that there are no measurements and no data of triple junction mobility, we cannot even estimate whether the ratio *m*TJ/*m*GB is finite. On the other hand, in the course of triple junction motion the straight grain boundary (Fig. 2, GB III) has to be extended. The velocity of its formation is unknown, but the kinetics of it should depend on the structure and properties of the generated grain boundary. Insofar as the rate of formation of this boundary can be interpreted as the velocity of the triple junction, which is proportional to its mobility according to equation (5). In the following a boundary system as shown in Fig. 2 with two identical curved boundaries (GB I and II) and a di€erent straight boundary (GB III) will be considered. The respective surface tensions and the mobilities of the boundaries are:

$$
\sigma 1 = \sigma 2 \equiv \sigma \neq \sigma 3, m\_1^{GB} = m\_2^{GB} \equiv m^{GB} \neq m\_3^{GB} \tag{6}
$$

In this case the shape of a steadily moving boundary system can be expressed by the equation

$$\frac{d^2\mathbf{y}}{dx^2} = -\frac{\nu}{m^{GB}\sigma} \frac{d\mathbf{y}}{d\mathbf{x}} \left[\mathbf{1} + \left(\frac{d\mathbf{y}}{dx}\right)^2\right] \tag{7}$$

with the boundary conditions

$$
\langle \chi(0) = 0, \chi(\infty) = \frac{a}{2}, \chi'(0) = \tan \theta \tag{8}
$$

as obvious from Fig. 2. Equations (6)-(8) completely define the problem. The shape of the stationary moving grain boundaries GB I and II (Fig. 1) is given by:

$$\text{hyp}(\mathbf{x}) = \xi \arccos\left(e^{-\frac{\mathbf{x}}{\xi} + \mathbf{C}\_1}\right) + \mathbf{C}\_2, \ \xi = \frac{a}{2\theta} \tag{9}$$

$$\mathcal{L}\_1 = \frac{1}{2} \ln(\sin \theta)^2,\\ \mathcal{C}\_2 = \,\,\xi \left(\frac{\pi}{2} - \theta\right) \tag{10}$$

The steady-state velocity of GB I and II is

Two-Step Sintering Applied to Ceramics 431

the kinetics of stress induced motion were different from the migration kinetics of curvature driven boundaries. Washburn, et al. 1952 and Li, et al., 1953 investigated planar low-angle boundaries in Zn under the influence of an external shear stress and observed the motion with polarized light in an optical microscope. Symmetrical low angle tilt boundaries consist of periodic arrangements of a single sets of edge dislocations. An external shear stress perpendicular to the boundary plane will cause a force on each dislocation and in summary a driving force on the boundary. The samples were exposed to a shear stress ranging from 10−1 to 10−3MPa. In aluminum (purity 99.999%) the yield stress is 15–20MPa, hence the applied shear stress is definitely in the elastic range. High angle symmetrical tilt boundaries also can be formally described as an arrangement of a single set of edge dislocations except that the dislocation cores overlap and the identity of the dislocations gets lost in the relaxed boundary structure. I showed that irrespective of the magnitude of the angle of rotation, grain boundaries can be moved under the action of the applied shear stress. The transition from low- to high-angle grain boundaries is revealed by a conspicuous step of the activation enthalpy at a misorientation angle of 13.6°. This holds for low angle as well as for high angle symmetrical tilt boundaries. For the curvature driven grain boundaries our results are in good agreement with previous experimental data [14] and one can see a strong dependency of the activation enthalpy on the misorientation angle, i.e. on the grain boundary structure. There is also a clear difference between the activation enthalpies for the stress induced motion of the planar high angle grain boundaries and the curvature driven migration of the curved high angle grain boundaries. Obviously, a dislocation in a high angle grain boundary does not relax completely its strain field and correspondingly, a biased elastic energy density induced by an applied shear stress will induce a force on all dislocations that comprise the grain boundary. The results prove that grain boundaries can be driven by an applied shear stress irrespective whether low- or high-angle boundaries. Obviously, the motion of the grain boundary is caused by the movement of the dislocations, which compose the grain boundary. The motion of an edge dislocation in a FCC crystal in reaction to an applied shear stress ought to be purely mechanical and not thermally activated. Obviously, the observed grain boundary motion is a thermally activated process controlled by diffusion. To understand this, one has to recognize first that grain boundary motion is a drift motion since it experiences a driving force that is smaller compared with thermal energy. Moreover, real boundaries are never perfect symmetrical tilt boundaries but always contain structural dislocations of other Burgers vectors. These dislocations have to be displaced by nonconservative motion to make the entire boundary migrate. The climb process requires diffusion, which can only be volume diffusion for low angle grain boundaries but grain boundary diffusion for high angle grain boundaries according to the observed activation enthalpies. The different behavior of curvature driven grain boundaries is not due to the curvature of the boundaries rather than due to a different effect of the respective driving force. While an applied shear stress couples with the dislocation content of the boundary in a curved grain boundary each individual atom experiences a drift

pressure to move in order to reduce curvature.

**6. Shape of the moving grain boundaries** 

The principal parameter which controls the motion of a grain boundary is the grain boundary mobility. In practically all relevant cases the motion of a straight grain boundary is the exception rather than the rule. That is why the shape of a moving grain boundary is of

$$\nu^{GB} = \frac{2\theta m^{GB}\sigma}{a} \tag{11}$$

The velocity of the triple junction *v*TJ can be expressed as (Soraes, et al., 1941), ( Galina, et al., 1987), (Fradkov, et al., 1988):

$$\mathbf{w}^{T\mathbf{j}} = \mathbf{m}^{T\mathbf{j}} \,\,\boldsymbol{\Sigma}\,\sigma\_{\mathbf{l}}\overrightarrow{\mathbf{r}\_{\mathbf{l}}}\tag{12}$$

where every �� ��� is the unit vector normal to the triple line and aligned with the plane of the adjacent boundary. If the angles at the triple junction are in equilibrium, the driving force is equal to zero and for a finite triple junction mobility the velocity *v*TJ should vanish as well. Consequently, for a finite *m*TJ, the motion of the triple junction disturbs the equilibrium of the angles and, as a result, drags the motion of the boundaries. For the situation given in Fig. 2.

$$\mathbf{v}^{\mathsf{T}f} = \mathsf{m}^{\mathsf{T}f} \{ 2\sigma \cos \theta - \sigma\_3 \} \tag{13}$$

In the case of steady-state motion of the entire boundary system the velocity of the triple junction equals the velocity of the grain boundaries. Therefore, the steady-state value of the angle θ is determined by equations (11) and (12):

$$\frac{2\theta}{2\cos\theta - \frac{\sigma\_3}{\sigma}} = \frac{m^{T/a}}{m^{GB}} = \Lambda \tag{14}$$

The dimensionless criterion Λ reflects the drag influence of the triple junction on the migration of the system. One can distinguish two limiting cases:

Λ → 0: In this case the angle θ tends to zero, i.e. the motion of the entire boundary system is governed by the mobility of the triple junction and the corresponding driving force. For the limit θ= 0º the velocity of the system is given with equation (13) by

$$\nu = m^{\text{T}} (2\sigma - \sigma\_3) \tag{15}$$

Λ →∞: In this case the angle θ tends to the value of thermodynamic equilibrium:

$$
\theta = \arccos\left(\frac{\sigma\_3}{2\sigma}\right) = \theta\_{eq.}\tag{16}
$$

The motion of the system is independent of the triple junction mobility and is governed only by the grain boundary mobility and the corresponding driving force. The velocity of the boundary system in this case with equations (11) and (16) is given by:

$$\nu = \frac{2\theta\_{eq}m\_{GB\sigma}}{a} \tag{17}$$

The two states of motion of the entire grain boundary system can be distinguished experimentally for a known ratio *σ*3/*σ* by measuring the contact angle θ.

#### **5. Influence of external shear stresses on grain boundary migration**

A method to activate and investigate the migration of planar, symmetrical tilt boundaries is influenced by external shear stress. It is shown that low- as well as high-angle boundaries could be moved by this shear stress. From the activation parameters for grain boundary migration, the transition from low- to high-angle boundaries can be determined. The migration kinetics were compared with results on curved boundaries, and it was shown that

��� <sup>=</sup> ������

The velocity of the triple junction *v*TJ can be expressed as (Soraes, et al., 1941), ( Galina, et al.,

where every �� ��� is the unit vector normal to the triple line and aligned with the plane of the adjacent boundary. If the angles at the triple junction are in equilibrium, the driving force is equal to zero and for a finite triple junction mobility the velocity *v*TJ should vanish as well. Consequently, for a finite *m*TJ, the motion of the triple junction disturbs the equilibrium of the angles and, as a result, drags the motion of the boundaries. For the situation given in Fig. 2.

In the case of steady-state motion of the entire boundary system the velocity of the triple junction equals the velocity of the grain boundaries. Therefore, the steady-state value of the

<sup>=</sup> ����

The dimensionless criterion Λ reflects the drag influence of the triple junction on the

Λ → 0: In this case the angle θ tends to zero, i.e. the motion of the entire boundary system is governed by the mobility of the triple junction and the corresponding driving force. For the

�� � ��� ���� �

Λ →∞: In this case the angle θ tends to the value of thermodynamic equilibrium:

� = ������ ���

� =

**5. Influence of external shear stresses on grain boundary migration** 

The motion of the system is independent of the triple junction mobility and is governed only by the grain boundary mobility and the corresponding driving force. The velocity of the

��������

The two states of motion of the entire grain boundary system can be distinguished

A method to activate and investigate the migration of planar, symmetrical tilt boundaries is influenced by external shear stress. It is shown that low- as well as high-angle boundaries could be moved by this shear stress. From the activation parameters for grain boundary migration, the transition from low- to high-angle boundaries can be determined. The migration kinetics were compared with results on curved boundaries, and it was shown that

migration of the system. One can distinguish two limiting cases:

limit θ= 0º the velocity of the system is given with equation (13) by

boundary system in this case with equations (11) and (16) is given by:

experimentally for a known ratio *σ*3/*σ* by measuring the contact angle θ.

1987), (Fradkov, et al., 1988):

angle θ is determined by equations (11) and (12):

� (11)

��� = ��� ∑ ���� ��� (12)

��� = ������ ��� � � ��� (13)

�=������ � ��� (15)

��� = Λ (14)

���=���� (16)

� (17)

the kinetics of stress induced motion were different from the migration kinetics of curvature driven boundaries. Washburn, et al. 1952 and Li, et al., 1953 investigated planar low-angle boundaries in Zn under the influence of an external shear stress and observed the motion with polarized light in an optical microscope. Symmetrical low angle tilt boundaries consist of periodic arrangements of a single sets of edge dislocations. An external shear stress perpendicular to the boundary plane will cause a force on each dislocation and in summary a driving force on the boundary. The samples were exposed to a shear stress ranging from 10−1 to 10−3MPa. In aluminum (purity 99.999%) the yield stress is 15–20MPa, hence the applied shear stress is definitely in the elastic range. High angle symmetrical tilt boundaries also can be formally described as an arrangement of a single set of edge dislocations except that the dislocation cores overlap and the identity of the dislocations gets lost in the relaxed boundary structure. I showed that irrespective of the magnitude of the angle of rotation, grain boundaries can be moved under the action of the applied shear stress. The transition from low- to high-angle grain boundaries is revealed by a conspicuous step of the activation enthalpy at a misorientation angle of 13.6°. This holds for low angle as well as for high angle symmetrical tilt boundaries. For the curvature driven grain boundaries our results are in good agreement with previous experimental data [14] and one can see a strong dependency of the activation enthalpy on the misorientation angle, i.e. on the grain boundary structure. There is also a clear difference between the activation enthalpies for the stress induced motion of the planar high angle grain boundaries and the curvature driven migration of the curved high angle grain boundaries. Obviously, a dislocation in a high angle grain boundary does not relax completely its strain field and correspondingly, a biased elastic energy density induced by an applied shear stress will induce a force on all dislocations that comprise the grain boundary. The results prove that grain boundaries can be driven by an applied shear stress irrespective whether low- or high-angle boundaries. Obviously, the motion of the grain boundary is caused by the movement of the dislocations, which compose the grain boundary. The motion of an edge dislocation in a FCC crystal in reaction to an applied shear stress ought to be purely mechanical and not thermally activated. Obviously, the observed grain boundary motion is a thermally activated process controlled by diffusion. To understand this, one has to recognize first that grain boundary motion is a drift motion since it experiences a driving force that is smaller compared with thermal energy. Moreover, real boundaries are never perfect symmetrical tilt boundaries but always contain structural dislocations of other Burgers vectors. These dislocations have to be displaced by nonconservative motion to make the entire boundary migrate. The climb process requires diffusion, which can only be volume diffusion for low angle grain boundaries but grain boundary diffusion for high angle grain boundaries according to the observed activation enthalpies. The different behavior of curvature driven grain boundaries is not due to the curvature of the boundaries rather than due to a different effect of the respective driving force. While an applied shear stress couples with the dislocation content of the boundary in a curved grain boundary each individual atom experiences a drift pressure to move in order to reduce curvature.

#### **6. Shape of the moving grain boundaries**

The principal parameter which controls the motion of a grain boundary is the grain boundary mobility. In practically all relevant cases the motion of a straight grain boundary is the exception rather than the rule. That is why the shape of a moving grain boundary is of

$$y(\mathbf{x}) = \{- (b\_F - b\_L) \arccos\left(\frac{\sin\Theta}{e^{\mathbf{x}^\*/b\_F}}\right) + \frac{a}{2} - b\_L \frac{\pi}{2} + b\_F \arccos\left(e^{b\_F \ln(\sin\Theta) - \mathbf{x}/b\_F}\right)\}\tag{18}$$

$$0 \le \mathbf{x} \le \mathbf{x}^\* \frac{a}{2} - b\_L \frac{\pi}{2} + b\_L \arccos\left(e^{b\_L \ln(\sin\Theta) - \mathbf{x}^\* ((b\_L/b\_F) - 1) - \mathbf{x}/b\_L}\right)\tag{18}$$

$$\mathbf{x} \ge \mathbf{x}^\*$$

$$\mathbf{b}\_{L} = \frac{\mathbf{b}\_{F} \{ \operatorname{arc} \cos \{ \sin \Theta / \mathbf{e}^{\mathbf{x}^{\*}/\mathbf{b}\_{F}} + \Theta - (\pi/2) - a/2 \} \}}{\operatorname{arc} \cos \{ \sin \Theta / \mathbf{e}^{\mathbf{x}^{\*}/\mathbf{b}\_{F}} \} - (\pi/2)} \mathbf{b} \tag{19}$$

Two-Step Sintering Applied to Ceramics 435

The authors wish to thank PRH-ANP, CAPES, LMCME-UFRN, Materials Laboratory-UFRN

Arzt, E., Ashby, M. F. & Verrall, R. A. (1993) Interface-controlled diffusional creep. Acta

Cannon, R. M., Rhodes, W. H. & Heuer, A. H. (1980) Plastic deformation of fine-grained alumina: I. interface-controlled diffusional creep. J. Am. Ceram. Soc. 63, 48-53 Chen, I.W. (1993). Mobility control of ceramic grain boundaries and interfaces, Materials

Chen, I.W. (2000). Grain boundary kinectics in oxide ceramics with the cubic fluorite crystal

Chen, I.W. & Wang, X.H. (2000). Sintering dense nanocrystaline ceramics without final-stage

Chen, I.W. & Xue, L.A. (1990). Development of superplastic structure ceramics. Journal of

Coble, R.L. (1965). Intermediate-stage sintering : Modification and correction of a lattice-

Czubayko, L., Sursaeva, V. G., Gottstein, G. & Shvindlerman, L. S., (1998) Influence of triple

Galina, A. V., Fradkov, V. E. and Shvindlerman, L. S., (1987) Physics Metals Metallogr., 63,

Herring, C. (1950). Effect of change of scale on sintering phenomena. Journal Applied to

Suppression of grain growth in sub-micrometer alumina via two-step sintering

microstructural characterization of two-step sintered ceria based electrolytes.

Herring, C. (1951). The physics of powder metallurgy. McGraw-Hill. New York. Pp. 143. Hesabi, Z.R. ; Haghighatzadeh, M. ; Mazaheri, M. ; Galusek, D.S.K. & Sadrnezhaad. (2008)

Jonghe, L.C. ; Rahaman, M.N. (2003). Sinterig ceramics. Handbook of Advanced Ceramics. Land, T. A., Martin, T. L., Potapenko, S., Palmore, G. T. & De Yoreo, J. J. (1999) Recovery of surfaces from impurity poisoning during crystal growth. Nature 399, 442±445. Lapa, C.M. ; Souza, D.P. ; Figueiredo, F.M.L. & Marques, F.M.B. (2009). Electrical and

Lapa, C.M. ; Souza, D.P. ; Figueiredo, F.M.L. & Marques, F.M.B. (2009). Two-step sintering ceria-based electrolytes. International Journal of Hydrogen Energy. Pp. 1-5. Li, C.H., Edwards, E.H., Washburn, J., Parker, E.R., (1954) Recent observations on the motion of small angle dislocation boundaries Acta Met. 2, 322–333.

structure and its derivates. Interface Science. Vol. 8. Pp. 147-156.

junctions on grain boundary motion. Acta Mater. 46, 5863±5871.

their coarse grain counterparts.

Metall. 31, 1977± 1989.

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American Ceramics Society. Vol. 73.

Chen, P.L. & Chen, I.W. (1997). Journal of American Ceramics Society.

method. Journal of the European Ceramic Society.

Journal of Power Sources. Vol. 187. Pp. 204-208.

grain growth. Nature. Vol. 404.

diffusion model. Vol. 36.

165.

Physics. Vol. 21.

**9. Acknowledgment** 

and NEPGN-UFRN.

**10. References** 

growth in the final stage of densification. Dense BaTiO3 ceramics with a grain size of 35 nm undergo distortions from cubic to various low-temperature ferroelectric structures. Dense fine grain Ni–Cu–Zn ferrite ceramics have the same saturation magnetization as

this assumption determines the fundamental concepts of grain structure evolution gives the Von Neumann–Mullins relation (Neumann, 1952 and Mullins, 1956). No doubt this relation forms the basis for practically all theoretical and experimental investigations as well as computer simulations of microstructure evolution in 2-D polycrystals in the course of grain growth. This relation is based on three essential assumptions, namely, (i) all grain boundaries possess equal mobilities and surface tensions, irrespective of their misorientation and crystallographic orientation of the boundaries; (ii) the mobility of a grain boundary is independent of its velocity; (iii) the third assumption relates directly to the triple junctions, namely, they do not affect grain boundary motion; therefore, the contact angles at triple junctions are in equilibrium and, due to the first assumption, are equal to 120°. As it was shown in (Neumann, 1952 and Mullins, 1956), for 2-D grain, the rate of change of the grain area *S* can be expressed by:

$$\frac{dS}{dt} = -A\_b \oint d\varphi \tag{20}$$

where *A*b=*m*b*σ*; *m*b being the grain boundary mobility, *σ* is the grain boundary surface tension. If the grain were bordered by a smooth line, the integral in Eq. (20) would equal 2*π*. However, owing to the discontinuous angular change at every triple junction, the angular interval Δφ=*π*/3 is subtracted from the total value of 2 *π* for each triple junction. Consequently:

$$\frac{dS}{dt} = -A\_b \left(2\pi - \frac{n\pi}{3}\right) = \frac{A\_b \pi}{3} (n - 6) \tag{21}$$

where *n* is the number of triple junctions for each respective grain, i.e. the topological class of the grain.

#### **8. Conclusions**

Many researchers have used the two-step sintering as a design process to obtain samples with a microstcture without grain growth in final stage of sintering. Same exemples that we can cite are:


growth in the final stage of densification. Dense BaTiO3 ceramics with a grain size of 35 nm undergo distortions from cubic to various low-temperature ferroelectric structures. Dense fine grain Ni–Cu–Zn ferrite ceramics have the same saturation magnetization as their coarse grain counterparts.

#### **9. Acknowledgment**

The authors wish to thank PRH-ANP, CAPES, LMCME-UFRN, Materials Laboratory-UFRN and NEPGN-UFRN.

#### **10. References**

434 Sintering of Ceramics – New Emerging Techniques

this assumption determines the fundamental concepts of grain structure evolution gives the Von Neumann–Mullins relation (Neumann, 1952 and Mullins, 1956). No doubt this relation forms the basis for practically all theoretical and experimental investigations as well as computer simulations of microstructure evolution in 2-D polycrystals in the course of grain growth. This relation is based on three essential assumptions, namely, (i) all grain boundaries possess equal mobilities and surface tensions, irrespective of their misorientation and crystallographic orientation of the boundaries; (ii) the mobility of a grain boundary is independent of its velocity; (iii) the third assumption relates directly to the triple junctions, namely, they do not affect grain boundary motion; therefore, the contact angles at triple junctions are in equilibrium and, due to the first assumption, are equal to 120°. As it was shown in (Neumann, 1952 and Mullins, 1956), for 2-D grain, the rate of

> �� ��

� ��� ��� � ��

where *A*b=*m*b*σ*; *m*b being the grain boundary mobility, *σ* is the grain boundary surface tension. If the grain were bordered by a smooth line, the integral in Eq. (20) would equal 2*π*. However, owing to the discontinuous angular change at every triple junction, the angular interval Δφ=*π*/3 is subtracted from the total value of 2 *π* for each triple junction.

where *n* is the number of triple junctions for each respective grain, i.e. the topological class

Many researchers have used the two-step sintering as a design process to obtain samples with a microstcture without grain growth in final stage of sintering. Same exemples that we

• Chen, I.W. & Wang, X.H. (2000) obtained samples of the Y2O3 with a grain size of 60nm can be prepared by a simple two-step sintering method, at temperatures of about 1,000ºC without applied pressure. The suppression of the final-stage grain growth is achieved by exploiting the difference in kinetics between grain boundary diffusion and grain-boundary migration. Such a process should facilitate the cost-effective

• 2: Lapa, et al., (2009) prepared samples of the yttrium and gadolinium-doped ceriabased electrolytes (20 at% dopant cation) with and without small Ga2O3-additions (0.5 mol%). The average grain sizes in the range 150–250 nm and densifications up to about 94% were found dependent on the sintering profile and presence of Ga. The grain boundary arcs in the impedance spectra increased significantly with Ga-doping, cancelling the apparently positive role of Ga on bulk transport, evidenced mostly in the

• 3: Wang, et al. (2006) used two-step sintering to sinter BaTiO3 and Ni–Cu–Zn ferrite ceramics to high density with unprecedentedly fine grain size, by suppressing grain

preparation of other nanocrystalline materials for practical applications.

� � � ���

� ��� ∮ �� (20)

� ����� (21)

change of the grain area *S* can be expressed by:

case of yttrium-doped materials.

�� ��

Consequently:

of the grain.

can cite are:

**8. Conclusions** 


**20** 

*México* 

**Ba1-XSrXTiO3 Ceramics Synthesized by an** 

**Alternative Solid-State Reaction Route** 

*Centro de Investigación y de Estudios Avanzados del IPN, Unidad Querétaro, Libramiento Norponiente No. 2000,* 

*Fracc. Real de Juriquilla, CP Querétaro, Qro.* 

R.A. Vargas-Ortíz, F.J. Espinoza-Beltrán and J. Muñoz-Saldaña

All materials respond to stimulus, whether it be an electric field, mechanical stress, heat or light. The manner and degree to which they respond varies and is often what determines which material is selected for a given application. On the most basic level, elastic materials deform in response to mechanical stress and return to their original form when the load is removed. Other materials conduct electricity in response to an applied voltage. Both of these are well-known phenomena, and materials with such behaviors are sometimes called "trivial". On the other hand are pyroelectric and piezoelectric materials, which generate an electric field with a stimulus of heat or mechanical stress, respectively (unexpected phenomenon) and are called "smart" or "functional" materials. Ferroelectric materials are materials that exhibit piezoelectricity and pyroelectricity, as well as the phenomenon which

Due to their unique properties, ferroelectric materials are widely used in all areas of electronics and microelectronics, such as cellular phones, computers, cars, airplanes and satellites [KENJI, BUCHANAN]. They have a high discharge dielectric constant (ε) [SHEPARD, RADHESHYAM ,ZHIN], which allows them to be used in high permitivity dielectric devices. Their pyroelectric behaviour is used in heat sensors [PADMAJA, YOO, WHATMORE], and their piezoelectricity is applied in devices like resonadores, sonars, horns, and actuators [GURURAJA, YAMASHITA 1997, YAMASHITA 1998, CHEN]. A combination of their properites are applied in electro-optical devices such as controlable diffraction grids, waveguides, etc. [HEIHACHI, HAMMER, BLOMQVIST]. They are also used in dynamic random access memory (DRAM) [KINGON 2000, KINGON 2006,

Many "novel" materials known today were developed many decades ago. Ferroelectric materials are no exception, having been discovered more than seven decades ago. Valasek reported the first ferroelectric material, Rochelle salt (potassium or sodium tetrahydrate tartatre, KNaC4H4O6 • 4H2O) in 1921 [VALASEK]. Subsequently, potassium dihydrogen phosphate (KH2PO4) was identified by Busch and Scherrer in 1935 [BUSCH], and barium titanate (BaTiO3 or BT) was noted for its unusual dielectric properties by Wainer and

KOTECKI] and non-volatile memory (NVRAM) [MASUI, KOHLSTEDT].

**1. Introduction** 

gives them their name (ferroelectricity).

