Intermediate sintering model

TABLE 16.6 Intermediate Stage Sintering Models Parameters fiar Simplified Models Represented by the Equation = K[(.y ClD)nR BTmf- t)... 

It is rather unfortunate that of ail the sintering stages the most important is also the most difficult to model. For example, any intermediate-stage model that does not take into account the details of particle packing has very limited validity. A cursory examination of the results shown in Fig. 10.10 should make this point amply clear. 

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... 

This concept can also be applied to the intermediate stage of sintering. Coble s intermediate stage model also took the surface area of the pore into account. The kinetic equations based on the two models are, in fact, the same except for the numerical constants. 

As in the intermediate stage, the uniform pore geometry assumed in the models precludes the occurrence of nondensifying mechanisms. Final stage sintering models have been developed by Coble (10) and Coleman and Deere (24). 

Coble did not specifically adapt the intermediate and final stage sintering models to account for hot pressing, but as described below, he used a different approach based on a modification of creep equations. However, as we shall show later, if the differential equations for matter transport are formulated with the proper dependence of the chemical potential on surface curvature and applied... 

The model of clusters or ensembles of sites and bonds (secondary supramolecular structure), whose size and structure are determined on the scale of a process under consideration. At this level, the local values of coordination numbers of the lattices of pores and particles, that is, number of bonds per one site, morphology of clusters, etc. are important. Examples of the problems at this level are capillary condensation or, in a general case, distribution of the condensed phase, entered into the porous space with limited filling of the pore volume, intermediate stages of sintering, drying, etc. 

R.L. Coble. Sintering crystalline solids I. Intermediate and final state diffusion models. J. Appl. Phys., 32(5) 787, 1961. 

As mentioned above, an area in which the concepts and techniques of statistical physics of disordered media have found useful application is the phenomenon of catalyst deactivation. Deactivation is typically caused by a chemical species, which adsorbs on and poisons the catalyst s surface and frequently blocks its porous structure. One finds that often reactants, products and reaction intermediates, as well as various reactant stream impurities, also serve as poisons and/or poison precursors. In addition to the above mode of deactivation, usually called chemical deactivation (2 3.), catalyst particles also deactivate due to thermal and mechanical causes. Thermal deactivation (sintering), in particular, and particle attrition and break-up due to thermal and mechanical causes, are amenable to modeling using the concepts of statistical physics of disordered media, but as already mentioned above the subject will not be dealt with in this paper. 

Since the introduction of a mathematical model for sintering by Kuczynski [12] numerous other models have been proposed. Reviews of these sintering kinetic models are given in references [13—19]. This description of sintering kinetics is organized into initial, intermediate, and final stage kinetic models. 

Diffusion models have been developed by Coble (1970) for initial, intermediate and final stages of pressure sintering including both the applied pressure and the surface energy as driving forces. The Mackenzie-Shuttleworth model is considered suitable for Newtonian viscous materials. 

Figure 10.14 (a) Tetrakaidecahedron model of intermediate-stage sintering, b) Expanded view of one of the cylindrical pore channels. The vacancies can diffuse down the grain boundary (dashed arrow) or through the bulk (solid arrows). Note that in both cases the vacancies are annihilated at the grain boundaries. 

Coble R L 1961 Sintering crystalline solids. I, intermediate and final state diffusion models J. Appl. Phys. 32 787-92... 

Creep equations can be appropriately modified as models of intermediate and final stages of sintering. For simplicity, the matter transport during creep of a dense solid is considered first. For a pure single crystal solid with cubic structure, which is a rod with a cross section of length L. Normal stresses pa nre applied to the rod on the sides, as shown in Fig. 5.23a. It is assumed that self-diffusion within the crystal... 

Coble RL (1961) Sintering crystalline solids. 1. Intermediate and final state diffusion model. J Appl Phys 32 787-793... 

Figure S.l. Coble s geometrical models for (a) intermediate stage and (b) final stage sintering.

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