1 Stages of Sintering

8.6.1 Stages of Sintering
Sintering is normally thought to occur in three sequential stages referred to as (1) the initial stage, (2) the intermediate stage, and (3) the final stage. In some analyses of sintering, an extra stage, stage 0, is considered which describes the instantaneous contact between the particles, when they are first brought together due to elastic deformation in response to surface energy reduction at the interface (9). However, we shall not consider this refinement. A stage represents an interval of time or density over which the microstructure is considered to be reasonably well defined. For polycrystalline materials, Fig. 8.8 shows the idealized geometrical structures that were suggested by Coble (10) as representative of the three
stages. For amorphous materials, the geometrical models assumed for the intermediate
and final stages are very different from those for the polycrystalline case and will be described appropriately later.

8.6.1.1 Initial Stage
The initial stage consists of fairly rapid interparticle neck growth by diffusion, vapor transport, plastic flow, or viscous flow. The large initial differences in surface curvature are removed in this stage, and shrinkage (or densification) accompanies neck growth for the densifying mechanisms. For a powder system consisting of spherical particles, the initial stage is represented as the transition between Figs. 8.8a and 8.8b. It is assumed to last until the radius of the neck between the particles has reached a value of _0.4–0.5 of the particle radius. For a powder system with an initial density of 0.5–0.6 of the theoretical density, this
corresponds to a linear shrinkage of 3 to 5% or an increase in density to _0.65 of the theoretical when the densifying mechanisms dominate.

8.6.1.2 Intermediate Stage
The intermediate stage begins when the pores have reached their equilibrium shapes as dictated by the surface and interfacial tensions (see Sec. 8.3). The pore phase is still continuous. In the sintering models, the structure is usually idealized in terms of a spaghetti-like array of porosity sitting along the grain edges as illustrated in Fig. 8.8c. Densification is assumed to occur by the pores simply shrinking to reduce their cross section. Eventually, the pores become unstable and pinch off, leaving isolated pores; this constitutes the beginning of the final stage. The intermediate stage normally covers the major part of the sintering process, and it is taken to end when the density is _0.9 of the theoretical.

8.6.1.3 Final Stage
The microstructure in the final stage can develop in a variety of ways, and we shall consider this in detail in Chapter 9. In one of the simplest descriptions, the final stage begins when the pores pinch off and become isolated at the grain corners, as shown by the idealized structure in Fig. 8.8d. In this simple description, the pores are assumed to shrink continuously and may disappear altogether. As outlined in Chapter 1, the removal of almost all of the porosity has been achieved in the sintering of several real powder systems. Some of the main parameters associated with the three idealized stages of sintering are summarized in Table 8.4, and examples of the microstructures (planar section) of real powder compacts in the initial, intermediate, and final stages are shown in Fig. 8.9


8.6.2 Modeling the Sintering Process
The analytical models, as outlined earlier, commonly assume that the particles in the initial powder compact are spherical and of the same size and that they are uniformly packed. With these assumptions, a unit of the powder system, called the geometrical model, can be isolated and analyzed. By imposing the appropriate boundary conditions, the remainder of the powder system can be considered as a continuum having the same macroscopic properties (e.g., shrinkage and densification rate) as the isolated unit. The derivation of the equations for the sintering kinetics follows a simple procedure: for the assumed geometrical model ,the mass transport equations are formulated and solved under the appropriate boundary conditions.

8.6.3 Initial Stage Models

8.6.3.1 Geometrical Parameters
The model for the initial stage consists of two equal-sized spheres in contact, referred to as the two-sphere model. Two slightly different geometries can be considered, depending on whether the mechanisms are nondensifying (Fig. 8.10a) or densifying (Fig. 8.10b). The two-sphere model for the densifying mechanisms accounts for interpenetration of the spheres (i.e., shrinkage) as well as neck growth. The neck formed between the particles is assumed to be circular with a radius X and with a surface having a circular cross section with a radius r. A
circular cross section for the neck surface is tantamount to assuming that the grain boundary energy is zero. The main geometrical parameters of the model are the principal radii of curvature of the neck surface r and X, the area of the neck surface A, and the volume of material transported into the neck V. These parameters are summarized in Fig. 8.10 and it is left as an exercise for the reader to derive them. It will be noticed that the parameters for the densifying model differ by only a small numerical factor compared to those for the nondensifying model











Figure 1. Schematic representations of a) a powder compact, b) nondensifying
neck growth, and c) densifying neck growth


Table I. Material Transport Mechanisms During Sintering.3
Method Transport Source Sink Densify
1 Surface Diffusion Surface Neck No
2 Lattice Diffusion Surface Neck No
3 Vapor Transport Surface Neck No
4 Boundary diffusion Grain Boundary Neck Yes
5 Lattice Diffusion Grain Boundary Neck Yes
6 Lattice Diffusion Dislocations Neck Yes


Barsoum defines solid-state sintering