**2. Mechanisms of two-step sintering**

During the process of the TSS the first step needs high temperature enough to achieve the critical diameter spherical (*dc*) of the core to become the crystallization, in this step the

Two-Step Sintering Applied to Ceramics 425

processing, fine starting powders, and low sintering temperatures may help to achieve submicron grain sizes in fully dense bodies. Such ceramics are suitable for kinetic studies of grain boundaries (Chen, 2000). The feasibility of densification without grain growth relies on the suppression of grain-boundary migration while keeping grain boundary diffusion active. Two-step sintering can be used to achieve a relative density of 98% by exploiting the ''kinetic window' that separates grain-boundary diffusion and grain-boundary migration. When conditions for two-step sintering fall below the ''kinetic window,'' a density \_ 96% cannot be achieved even if a starting density of 70% is achieved at T1, as grain growth may still be suppressed but densification will be exhausted. Above the ''kinetic window,'' grain growth is likely to occur (Wright, 2008). 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 preparation of other nanocrystalline materials for practical applications. To succeed in two-step sintering, a suficiently high starting density should be obtained during the first step. When the density is above 70%, porosimetry data have shown that all pores in Y2O3 become subcritical and unstable against shrinkage (which occurs by capillary action). These pores can be called as long as grain-boundary diffusion allows it, even if the particle network is frozen as it clearly is in the second step (Chen & Wang, 2000). From the thermodynamics aspect, at a temperature range where grain boundary diffusion is active, but grain boundary migration is sufficiently sluggish, densification would continue without any significant grain growth. On basis of this idea was developed to suppress the accelerated grain growth at the final stage of sintering by triple junctions. To take the advantage of boundary dragging by triple junctions, a critical density at first should be achieved where sufficient triple junctions exist throughout the body as pins. Then with decreasing the sintering temperature to a critical degree, the grain growth would be stopped by triple junctions while densification may not be impaired. In doing so, samples have to be exposed to prolonged isothermal heating at the second (low temperature) step. As in a TSS regime, the triple junctions are going to prohibit grain growth, while unstable pores can shrink with low temperature annealing, seemingly the source of different densities lies in the pore size and distribution which needs to be further investigated. Certainly, formation of inhomogeneous porosity due to the increased tendency of nanopowder to form agglomerates complicates the situation. To solve this problem, one can use larger particles with lower agglomeration degree and shape green bodies with advanced methods to obtain a more homogenous structure. Under this

condition, one can expect successful TSS at lower temperatures (Hesabi et al., 2008).

**3. Grain boundary kinetics during intermediate and final stage sintering** 

Sintering data are often used to infer the rate-controlling mechanism following the scaling

micrometer grain size (Lapa, 2009).

analysis of Herring (Herring, 1950 & Herring, 1951).

Sophisticated firing profiles are an alternative to compositional effects, to obtain dense ceramics with proper microstructure. Two-step sintering profiles, including optimized combinations of peak and dwell sintering temperatures, produced nanostructured materials with high densification at reasonably low temperatures, due to different grain growth and densification kinetics (Chen & Wang, 2000). Was observed that samples processed by twostep sintering show best results the density, around 94% of theoretical density and sub-

relative density (*ρ*) need be the same or higher than 75% of the teorical density to obtain unstable pores and the sintering the material be kept. In the second step is necessary the temperature to keep the densification until the end of the sintering, but avoiding the grain growth. An important reason to have an understanding of two-step sintering is the possibility to increase the heat rate of sintering, to avoid the grain growth and to obtain a material with improved mechanical, thermal, electrical and optical properties in the materials (Chen & Wang, 2000). The absence of grain growth in second-step sintering has important implications for kinetics. Grain coarsening creates a powerful dynamic that constantly refreshes the microstructure. Statistically, only one-eighth of all grains survive every time the size of the grains doubles. This evolution can be a source of enhanced kinetics. Even without grain growth, enhanced kinetics has also been suspected in cases when microstructure evolution is otherwise robust; for example, in fine-grain superplasticity (McFadden et al., 1999; Wakai et al., 1990 & Chen & Xue, 1990). Because second-step sintering proceeds in a ´frozen' microstructure, it should have slower kinetics. Yet the slower kinetics is sufficient for reaching full density, while providing the benefit of suppressing grain growth. The diffusion kinetics is quantified in the frozen microstructure by measuring the densification rates in the second step, and comparing them with the prediction based on Herring's dimensional argument (Herring, 1950) for normalized densification rate (dr/rdt, where r is relative density and t is time):

$$\frac{d\rho}{\rho dt} = F(\rho) \left( \frac{\chi \Omega}{GkT} \right) \left( \frac{\delta D}{G^3} \right) \tag{1}$$

Here *γ* is surface energy, *Ω* is atomic volume, *G* is grain size, *d* is grain-boundary thickness, and *D* is grain-boundary diffusivity. In the above, *γΩ*/GkT may be viewed as the normalized driving force, and *δD*/G3 is the standard kinetic factor that enters the strain-rate equation for grain-boundary processes such as sintering, diffusional creep and superplasticity. The remaining dimensionless prefactor on the right-hand side, F, is independent of the grain size as such, but could depend on other aspects of the microstructure such as density and pore distribution (Wei & Wang, 2000).

Grain boundaries in ceramics have been extensively investigated in recent years with the intent to understand their structures and mechanical/electrical properties. Grain boundaries are also important for kinetic phenomena, such as sintering, grain growth, diffusional creep and superplasticity. Their importance increases as the grain size decreases, since the ratio of grain boundary to the grain interior is inversely proportional to the grain size. In addition, this ratio also dictates that there is a large capillary pressure (and its variation) in fine grain materials. For a typical grain boundary energy (and surface energy) in ceramics of 1 J/m2, the capillary pressure is of the order of 2000 MPa at a grain size of 1 nm, 200 MPa at 10 nm, and 20MPa at 100 nm. These pressures are rather significant and may cause additional kinetic effects at intermediate and high temperatures. In general, the dominant kinetic paths in submicron powders and ceramic bodies are surfaces and grain boundaries, the latter becoming increasingly important as the relative density reaches toward 100%. "Clean" experiments on grain boundary kinetics without the "contamination" of surface effects can be undertaken provided full density is first achieved. Common ceramic firing processes, however, always induce rapid grain growth when the relative density exceeds 85%, because of the breakdown of pore channels at three grain junctions and the resulting reduction of the pore drag on grain boundary migration. Nevertheless, the combination of good powder

relative density (*ρ*) need be the same or higher than 75% of the teorical density to obtain unstable pores and the sintering the material be kept. In the second step is necessary the temperature to keep the densification until the end of the sintering, but avoiding the grain growth. An important reason to have an understanding of two-step sintering is the possibility to increase the heat rate of sintering, to avoid the grain growth and to obtain a material with improved mechanical, thermal, electrical and optical properties in the materials (Chen & Wang, 2000). The absence of grain growth in second-step sintering has important implications for kinetics. Grain coarsening creates a powerful dynamic that constantly refreshes the microstructure. Statistically, only one-eighth of all grains survive every time the size of the grains doubles. This evolution can be a source of enhanced kinetics. Even without grain growth, enhanced kinetics has also been suspected in cases when microstructure evolution is otherwise robust; for example, in fine-grain superplasticity (McFadden et al., 1999; Wakai et al., 1990 & Chen & Xue, 1990). Because second-step sintering proceeds in a ´frozen' microstructure, it should have slower kinetics. Yet the slower kinetics is sufficient for reaching full density, while providing the benefit of suppressing grain growth. The diffusion kinetics is quantified in the frozen microstructure by measuring the densification rates in the second step, and comparing them with the prediction based on Herring's dimensional argument (Herring, 1950) for normalized

densification rate (dr/rdt, where r is relative density and t is time):

ௗఘ ఘௗ௧

microstructure such as density and pore distribution (Wei & Wang, 2000).

ൌ ܨሺߩሻ <sup>ቀ</sup> ఊஐ

Here *γ* is surface energy, *Ω* is atomic volume, *G* is grain size, *d* is grain-boundary thickness, and *D* is grain-boundary diffusivity. In the above, *γΩ*/GkT may be viewed as the normalized driving force, and *δD*/G3 is the standard kinetic factor that enters the strain-rate equation for grain-boundary processes such as sintering, diffusional creep and superplasticity. The remaining dimensionless prefactor on the right-hand side, F, is independent of the grain size as such, but could depend on other aspects of the

Grain boundaries in ceramics have been extensively investigated in recent years with the intent to understand their structures and mechanical/electrical properties. Grain boundaries are also important for kinetic phenomena, such as sintering, grain growth, diffusional creep and superplasticity. Their importance increases as the grain size decreases, since the ratio of grain boundary to the grain interior is inversely proportional to the grain size. In addition, this ratio also dictates that there is a large capillary pressure (and its variation) in fine grain materials. For a typical grain boundary energy (and surface energy) in ceramics of 1 J/m2, the capillary pressure is of the order of 2000 MPa at a grain size of 1 nm, 200 MPa at 10 nm, and 20MPa at 100 nm. These pressures are rather significant and may cause additional kinetic effects at intermediate and high temperatures. In general, the dominant kinetic paths in submicron powders and ceramic bodies are surfaces and grain boundaries, the latter becoming increasingly important as the relative density reaches toward 100%. "Clean" experiments on grain boundary kinetics without the "contamination" of surface effects can be undertaken provided full density is first achieved. Common ceramic firing processes, however, always induce rapid grain growth when the relative density exceeds 85%, because of the breakdown of pore channels at three grain junctions and the resulting reduction of the pore drag on grain boundary migration. Nevertheless, the combination of good powder

ீ்ቁ ቀఋ

ீ<sup>య</sup> <sup>ቁ</sup> (1)

processing, fine starting powders, and low sintering temperatures may help to achieve submicron grain sizes in fully dense bodies. Such ceramics are suitable for kinetic studies of grain boundaries (Chen, 2000). The feasibility of densification without grain growth relies on the suppression of grain-boundary migration while keeping grain boundary diffusion active. Two-step sintering can be used to achieve a relative density of 98% by exploiting the ''kinetic window' that separates grain-boundary diffusion and grain-boundary migration. When conditions for two-step sintering fall below the ''kinetic window,'' a density \_ 96% cannot be achieved even if a starting density of 70% is achieved at T1, as grain growth may still be suppressed but densification will be exhausted. Above the ''kinetic window,'' grain growth is likely to occur (Wright, 2008). 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 preparation of other nanocrystalline materials for practical applications. To succeed in two-step sintering, a suficiently high starting density should be obtained during the first step. When the density is above 70%, porosimetry data have shown that all pores in Y2O3 become subcritical and unstable against shrinkage (which occurs by capillary action). These pores can be called as long as grain-boundary diffusion allows it, even if the particle network is frozen as it clearly is in the second step (Chen & Wang, 2000). From the thermodynamics aspect, at a temperature range where grain boundary diffusion is active, but grain boundary migration is sufficiently sluggish, densification would continue without any significant grain growth. On basis of this idea was developed to suppress the accelerated grain growth at the final stage of sintering by triple junctions. To take the advantage of boundary dragging by triple junctions, a critical density at first should be achieved where sufficient triple junctions exist throughout the body as pins. Then with decreasing the sintering temperature to a critical degree, the grain growth would be stopped by triple junctions while densification may not be impaired. In doing so, samples have to be exposed to prolonged isothermal heating at the second (low temperature) step. As in a TSS regime, the triple junctions are going to prohibit grain growth, while unstable pores can shrink with low temperature annealing, seemingly the source of different densities lies in the pore size and distribution which needs to be further investigated. Certainly, formation of inhomogeneous porosity due to the increased tendency of nanopowder to form agglomerates complicates the situation. To solve this problem, one can use larger particles with lower agglomeration degree and shape green bodies with advanced methods to obtain a more homogenous structure. Under this condition, one can expect successful TSS at lower temperatures (Hesabi et al., 2008).

Sophisticated firing profiles are an alternative to compositional effects, to obtain dense ceramics with proper microstructure. Two-step sintering profiles, including optimized combinations of peak and dwell sintering temperatures, produced nanostructured materials with high densification at reasonably low temperatures, due to different grain growth and densification kinetics (Chen & Wang, 2000). Was observed that samples processed by twostep sintering show best results the density, around 94% of theoretical density and submicrometer grain size (Lapa, 2009).

#### **3. Grain boundary kinetics during intermediate and final stage sintering**

Sintering data are often used to infer the rate-controlling mechanism following the scaling analysis of Herring (Herring, 1950 & Herring, 1951).

Two-Step Sintering Applied to Ceramics 427

Fig. 1. Schematic Arrhenius plot for grain boundary diffusion, mobility of pore/grainboundary junction or four-grain junction, and intrinsic mobility of grain boundary (without

Simple arguments then show that below a certain temperature, equilibrium grain boundary migration does not obtain since the boundaries are effectively pinned by the nodal points (lines). As mentioned above, enhanced grain boundary migration is often observed in superplastic deformation of fine grain oxides. The grain growth in this case is found to be controlled by the plastic strain. Indeed the ratio of grain size is essentially of the same order as the ratio of specimen dimensions before and after deformation (Chen & Xue, 1990). This may be regarded as opposite to the suppression of grain boundary migration described above. It is likely that in both cases, the dynamics of the nodal line/point are important. In superplasticity, the dynamics are enhanced to facilitate grain boundary migration. In low temperature sintering, the dynamics are inhibited to suppress grain boundary migration. A better knowledge of the structures of the grain boundary nodal points and lines, in both equilibrium configurations and in dynamic configurations would be required for a full understanding of the grain boundary kinetics. One interesting observation though is that a parallel effect of solute is seen in all three cases: normal grain growth, dynamic grain growth, and sintering without grain growth. For example, solutes that enhance normal grain growth also cause faster dynamic grain growth, and solutes that suppress normal grain growth likewise show a higher temperature *T*2 in the kinetic window for sintering without grain growth. Thus, while the kinetics of the nodal point/line may be distinct from that of grain boundary diffusion, they may not be entirely independent of each other. Recent studies of high-purity zinc have shown that grain-boundary migration can be severely hampered by the slow mobility of grain junctions at lower temperatures, the latter having a higher activation energy (Czubayko, et al., 1998). It is possible that a similar process, in which grain junctions as well as grain boundary/pore junctions impede grain-boundary migration, may here explain the apparent suppression of grain growth at lower temperatures. Interface kinetics in very fine grain polycrystals is sometimes limited due to

**1/T** 

extrinsic drag due to nodal points/lines) (Source: Gary J. Wright, 2008).

**Log rate** 

$$\frac{d\rho}{dt} = \left(\frac{f(\rho)}{kTR^m}\right) D\_0 \exp\{-Q/kT\} \tag{2}$$

In the above, *ρ* is the relative density, *m* is either 3 for lattice diffusion or 4 for grain boundary diffusion, *D*0 and *Q* refer to the pre-factor and activation energy of either lattice diffusion or grain boundary diffusion, and *f* is a constant that is dependent on the pore/grain geometry of the sintering body. Over a range of relative density, from 60% to 90%, some model calculations suggest that *f* is relatively constant (Coble, 1965 & Swinkels and Ashby, 1981). Thus, Eq. (2) can be used to deduce the diffusivity and the rate controlling mechanism if the densification rate and the grain size are known. In practice, plotting Log (*Tdρ=dt)Rm* against 1*/T* usually yields a straight line regardless whether *m* is chosen as 3 or 4. This is due to the relatively poor resolution of such plotting method, the unavoidable scatter of the data, and the uncertainty of the value of *f(ρ)*. Therefore, it is usually not possible to definitively state that the sintering mechanism is via grain boundary diffusion or lattice diffusion based on the scaling analysis alone. On the other hand, the inferred values of diffusivities, and especially those of the activation energy, are usually quite different depending on whether *m* is chosen as 3 or 4. Thus, when independent diffusivity data are available, e.g., from grain boundary mobility measurements, a self-consistency check may be applied to infer whether the lattice or grain boundary mechanism applies. Using this method, is possible to conclude that later stage sintering of submicron CeO2 and Y2O3 powders is controlled by grain boundary diffusion (Chen & Chen, 1997). As cited above, to achieve densification without grain growth, is necessary to first fire the ceramic at a higher temperature (*T*1) to a relative density of 75% or more, then sinter it at a lower temperature (*T*2) for an extended time to reach full density. This schedule is different from the conventional practice for sintering ceramics, in which the temperature always increases or, at least is held constant at the highest temperature, until densification is complete. The temperature *T*2 required for the second step decreased with the increasing grain size. However, if *T*2 is too low, then sintering proceeds for a while and then becomes exhausted. On the other hand, if *T*2 is too high, grain growth still occurs in the second step. Chen 2000, observed that the same shape in different Y2O3 but the medium temperature is shifted depending on the solute added. The results imply that grain boundaries previously stabilized at a higher temperature are difficult to migrate at a lower temperature even though they may still provide fast diffusion paths. The presence of impurity or solute segregation is not essential for this observation, since the same observation was found in pure Y2O3 as well as Y2O3 doped with both diffusion enhancing and diffusion-suppressing solutes. Therefore, the suppression of migration is not due to solute pinning. This suggests, for the first time that the mechanism for grain boundary migration is not grain boundary diffusion even in a pure substance. If the activation energy of the additional step is higher than that of grain boundary diffusion, it could explain why grain boundary migration is inhibited at low temperatures but not at high temperature. The most likely candidates for such step are movement of nodal points or nodal lines on the grain boundary, such as fourgrain junctions, pore-grain boundary junctions, or three-grain junctions. The structures of these nodal points and lines may be special and they could become stabilized by prior high temperature treatment, rendering them difficult to alter to accommodate the subsequent movement of migrating grain boundary at low temperatures. Empirically, this may be modeled by assigning mobility to the nodal point (line), whose ratio to grain boundary mobility decreases with increasing temperature.

In the above, *ρ* is the relative density, *m* is either 3 for lattice diffusion or 4 for grain boundary diffusion, *D*0 and *Q* refer to the pre-factor and activation energy of either lattice diffusion or grain boundary diffusion, and *f* is a constant that is dependent on the pore/grain geometry of the sintering body. Over a range of relative density, from 60% to 90%, some model calculations suggest that *f* is relatively constant (Coble, 1965 & Swinkels and Ashby, 1981). Thus, Eq. (2) can be used to deduce the diffusivity and the rate controlling mechanism if the densification rate and the grain size are known. In practice, plotting Log (*Tdρ=dt)Rm* against 1*/T* usually yields a straight line regardless whether *m* is chosen as 3 or 4. This is due to the relatively poor resolution of such plotting method, the unavoidable scatter of the data, and the uncertainty of the value of *f(ρ)*. Therefore, it is usually not possible to definitively state that the sintering mechanism is via grain boundary diffusion or lattice diffusion based on the scaling analysis alone. On the other hand, the inferred values of diffusivities, and especially those of the activation energy, are usually quite different depending on whether *m* is chosen as 3 or 4. Thus, when independent diffusivity data are available, e.g., from grain boundary mobility measurements, a self-consistency check may be applied to infer whether the lattice or grain boundary mechanism applies. Using this method, is possible to conclude that later stage sintering of submicron CeO2 and Y2O3 powders is controlled by grain boundary diffusion (Chen & Chen, 1997). As cited above, to achieve densification without grain growth, is necessary to first fire the ceramic at a higher temperature (*T*1) to a relative density of 75% or more, then sinter it at a lower temperature (*T*2) for an extended time to reach full density. This schedule is different from the conventional practice for sintering ceramics, in which the temperature always increases or, at least is held constant at the highest temperature, until densification is complete. The temperature *T*2 required for the second step decreased with the increasing grain size. However, if *T*2 is too low, then sintering proceeds for a while and then becomes exhausted. On the other hand, if *T*2 is too high, grain growth still occurs in the second step. Chen 2000, observed that the same shape in different Y2O3 but the medium temperature is shifted depending on the solute added. The results imply that grain boundaries previously stabilized at a higher temperature are difficult to migrate at a lower temperature even though they may still provide fast diffusion paths. The presence of impurity or solute segregation is not essential for this observation, since the same observation was found in pure Y2O3 as well as Y2O3 doped with both diffusion enhancing and diffusion-suppressing solutes. Therefore, the suppression of migration is not due to solute pinning. This suggests, for the first time that the mechanism for grain boundary migration is not grain boundary diffusion even in a pure substance. If the activation energy of the additional step is higher than that of grain boundary diffusion, it could explain why grain boundary migration is inhibited at low temperatures but not at high temperature. The most likely candidates for such step are movement of nodal points or nodal lines on the grain boundary, such as fourgrain junctions, pore-grain boundary junctions, or three-grain junctions. The structures of these nodal points and lines may be special and they could become stabilized by prior high temperature treatment, rendering them difficult to alter to accommodate the subsequent movement of migrating grain boundary at low temperatures. Empirically, this may be modeled by assigning mobility to the nodal point (line), whose ratio to grain boundary

்ோቁ ܦ݁ݔሺെܳȀ݇ܶሻ (2)

ௗఘ ௗ௧ ൌ ቀ ሺఘሻ

mobility decreases with increasing temperature.

Fig. 1. Schematic Arrhenius plot for grain boundary diffusion, mobility of pore/grainboundary junction or four-grain junction, and intrinsic mobility of grain boundary (without extrinsic drag due to nodal points/lines) (Source: Gary J. Wright, 2008).

Simple arguments then show that below a certain temperature, equilibrium grain boundary migration does not obtain since the boundaries are effectively pinned by the nodal points (lines). As mentioned above, enhanced grain boundary migration is often observed in superplastic deformation of fine grain oxides. The grain growth in this case is found to be controlled by the plastic strain. Indeed the ratio of grain size is essentially of the same order as the ratio of specimen dimensions before and after deformation (Chen & Xue, 1990). This may be regarded as opposite to the suppression of grain boundary migration described above. It is likely that in both cases, the dynamics of the nodal line/point are important. In superplasticity, the dynamics are enhanced to facilitate grain boundary migration. In low temperature sintering, the dynamics are inhibited to suppress grain boundary migration. A better knowledge of the structures of the grain boundary nodal points and lines, in both equilibrium configurations and in dynamic configurations would be required for a full understanding of the grain boundary kinetics. One interesting observation though is that a parallel effect of solute is seen in all three cases: normal grain growth, dynamic grain growth, and sintering without grain growth. For example, solutes that enhance normal grain growth also cause faster dynamic grain growth, and solutes that suppress normal grain growth likewise show a higher temperature *T*2 in the kinetic window for sintering without grain growth. Thus, while the kinetics of the nodal point/line may be distinct from that of grain boundary diffusion, they may not be entirely independent of each other. Recent studies of high-purity zinc have shown that grain-boundary migration can be severely hampered by the slow mobility of grain junctions at lower temperatures, the latter having a higher activation energy (Czubayko, et al., 1998). It is possible that a similar process, in which grain junctions as well as grain boundary/pore junctions impede grain-boundary migration, may here explain the apparent suppression of grain growth at lower temperatures. Interface kinetics in very fine grain polycrystals is sometimes limited due to

Two-Step Sintering Applied to Ceramics 429

features of the motion of this system can be established. (a) A steady-state motion of the

��� <sup>=</sup> ��

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

������

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

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:

�� = ��

� � ��

as obvious from Fig. 2. Equations (6)-(8) completely define the problem. The shape of the

� ���

������ ���� �� = � ��

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

���� �� �� �� � ���

��0� = 0� ��∞� <sup>=</sup> �

���� = ������� ����

�� � ��� � ��

��� �

����� � = �

� ��� ��� (3)

�� (4)

� = ���� (5)

�� (6)

� (7)

�� (9)

� − �� (10)

�0� = ��� � (8)

Λ = ����

� =

whole system is possible indeed. (b) The dimensionless criterion Λ:

determined by the grain boundary mobility and the acting driving force:

�� = �� � � � ��� ��

��� ��� = − �

stationary moving grain boundaries GB I and II (Fig. 1) is given by:

�� <sup>=</sup> � �

mobility and the grain boundary surface tension *σ*:

equation

with the boundary conditions

The steady-state velocity of GB I and II is

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 overall kinetics.
