Sintering equation

S. Pejovnik et al., Statistical analysis of the validity of sintering equations . Powder Metallurgy International, 11 (1), 22-3 (1979). 

Fig. 5.18 a Schematic of the hexagonal neck for intermediate stage sintering equations of polycrystalline ceramics, b The section shows a cut through the neck with the atomic flux paths for grain boundary and lattice diffusion. Reproduced with permission from [1]. 

The sintering equations are commonly expressed with densification rates. Porosity P and relative density p are related by P = 1 — p, therefore, by differentiating Eq. (5.163) with respect to time, there is... 

Similar assumptions are made for the final sintering stage, so that nondensifying mechanisms can be neglected, which is similar to that in the intermediate stage model [27, 38]. The procedure to derive the sintering equations for the intermediate... 

There are also various empirical sintering equations that can be found in the open literature. One of them is as follows ... 

This equation indicates that, at the initial stage, AC for hot pressing is similar to that for normal sintering, but ysv + is used instead of ysv- Because and a are constant, hot pressing equations can be derived from the sintering equations by simply changing ysv into ysv + Pi aln. 

Besides those discussed above, the sintering equations can be derived in an alternative way by solving the differential equations of the atomic flux that is subject to appropriate boundary conditions [70]. 

A simple geometrical model consisting of spherical particles with a diameter a, arranged as a simple cubic pattern, as shown in Fig. 5.26a, is used to derive the sintering equations for the mechanism of grain boundary diffusion [70]. It is... 

The sintering equation for lattice diffusion can be derived in a similar way. The linear densification strain rate can also be derived from Eq. (5.216) by replacing rPgb(5gb with 2XDy, where D is the lattice diffusion coefhcient [71]. Because there is 0 = a l(nX ), the linear densification strain rate is given by... 

The exponent a in the scaling law can also be deduced from the sintering equations in Table 4.2 without following Herring s derivation. The sintering... 

All of the sintering equations were derived under the assumption that a local equiUbrium of atoms with capillary pressure is maintained everywhere, in the atom source and in the atom sink. (This assumption is acceptable.) The dihedral angle of 180° is also an acceptable assumption because the dihedral angle affects only numerical constants in kinetic equations, and not the sintering variables (Eq. (4.7)). 

Figure 8.10 Geometrical parameters for the two-sphere model used in the derivation of the initial stage sintering equations for crystalhne particles. The geometries shown correspond to those for (a) the nondensifying mechanisms and (h) the densifying mechanisms.

Summary and Limitations of the Initial Stage Sintering Equations... 

The derivation of the sintering equations for Scherer s model is quite similar to that outlined earlier for Frenkel s initial stage model. The result is (21)... 

A final stage sintering equation for grain boundary diffusion was not derived by Coble, but apart from a difference in the numerical constant, the equation may be expected to be identical to Eq. (8.49). As we shall see later in Sec. 8.10, Coble later developed models for diffusional sintering with an applied pressure from... 

The derivation of the sintering equation for lattice diffusion follows a procedure similar to that described above for grain boundary diffusion. The linear densification strain rate can also be obtained from Eq. (8.116) by replacing irDgbSgb with IXDi, where Di is the lattice diffusion coefficient (65). Remembering that 4> = a /( aX ), these substitutions give... 

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