Sintering Frenkel model

Frenkel Model - A model developed by Russian scientist Frenkel to describe coalescence/sinter-ing of particles of metals and plastics. See the following references for more information Mascia, L., Thermoplastics, Matena/5 Eng., 2 ed., Elsevier Applied Science, New York (1989) and Lontz,J. F., in Fundamental Phenomena in the Material Sciences, Vol. 1 (L. J. Borris and H. H. Hansner, eds.), p. 37, Plenum Press, New York (1964). 

The model of viscous sintering was developed by Frenkel (24), who derived the following expression for the rate of coalescence of spherical adjacent particles ... 

The coalescence of polymers is driven by the work of surface tension, which counteracts the viscous dissipation associated with the molecular diffusion within the coalescing domain. This phenomenon is often referred to in the literature as polymer sintering. In the rotational molding process, coalescence occurs at temperatures above that of the material melting point when dealing with semicrystalline polymers, or above the glass transition temperature for amorphous resins. The first analytical model describing the coalescence process was proposed by Frenkel ... 

Fig. 5.16 Geometrical parameters of the two-sphere model used in the derivation of the initial stage equation for viscous sintering by Frenkel. Reproduced with permission from [1]. Copyright 2003, CRC Press...

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 feature that arises from our consideration of the analytical models is the difference in complexity between the sintering phenomena in polycrystalline materials and amorphous materials. The analysis of viscous sintering on the basis of Frenkel s energy balance concept appears relatively simple in principle. The idealization of the structure of amorphous materials leads to analytical solutions that describe the sintering behavior in a very satisfactory manner. 

Flow fields for viscous sintering of spheres (schematic). Right-hand side, uniaxial contraction assumed in Frenkel s model. Left-hand side, form expected in real situations. From Uhlmann el al. (18]. Note added in proof Recent simulations using finite element analysis (A. Jagota and P.R. Dawson, to be published in J. Am. Ceram. Soc.) show that the flow field on the right is, indeed, correct. 

Theories of Viscous Sintering describes the concept of energy balance by which Frenkel first analyzed sintering, and it reviews the microstructural models that have been developed. Effects of a distribution of pore sizes and of gas trapped in the pores are considered. 

Figure 9. Frenkel s model for early-stage sintering viscous flow [41].

Many aspects can be studied from the kinetic equations, as densification rate, determination of sintering mechanisms, and activation energy. The equation developed by Coble allows to estimate the sintering mechanisms for the initial stage, based on the two spheres Frenkel s model, as indicated in the Eq. 15 [43] ... 

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