Grain shape equilibrium

The needle-like grain shape seems to be kinetically determined and therefore not the equilibrium crystal shape [30,38], The estimated equilibrium aspect ratio (ratio of grain length to thickness) is 1.3, not as high as 20, as was found to be the case for the silicon nitride materials. The microstructural formation must therefore be explained on the basis of growth mechanisms. 

An alloy is cooled from a temperature at which it has a single-phase structure (a) to a temperature at which the equilibrium structure is two-phase (a -i- ji). During cooling, small precipitates of the P phase nucleate heterogeneously at a grain boundaries. The nuclei are lens-shaped as shown below. 

Grain boundaries (and boundaries between phases) are elements of the microstructure of crystalline solids, being characterized by their number, shape, and topological arrangement. The microstructure is a non-equilibrium property. In the next section we discuss grain boundaries. 

If surface diffusion or vapor transport is rapid enough, the pores will maintain their quasi-static equilibrium shape, illustrated in Fig. 16.1 in the form of four cylindrical sections of radius R.6 The dihedral angle at the four intersections with grain boundaries, ip, will obey Young s equation, ip is related to 6 by sin( /j/2) = cos 0. 

Figure 16.14 Cross section of equilibrium shape of pore on grain boundary as in...

We wish to prove by means of the Wulff construction (Section C.3.1) that the equilibrium shape of the grain boundary nucleus in Fig. 19.12 is indeed composed of two spherical-cap-shaped interfaces. 

Schaefer et al. have developed a somewhat different method which involves measuring the shape and depth of grooves, which are formed at equilibrium when a liquid meets a grain boundary in the solid, with a small temperature gradient imposed. The curvature at each point is related to the local temperature and to the surface free energy. Measurements were... 

When refering to shape selectivity properties related to diffusivity, it seems obvious that the larger the zeolite grain, the higher will be the volume/sur f ace ratios and the shape selectivity, since the reaction will be more diffusion controlled. The external surface area represents different percents of the total zeolite area depending on the size of the grains which could be important if the active sites at the external surface also play a role in the selectivity. For instance in the case of toluene alkylation by methanol, the external surface acid sites will favor the thermodynamical equilibrium due to isomerization reactions (o m p-xylene - 25 50 25 at 400 C) while diffusivity resistance will favor the less bulky isomer namely the para-xylene. It may therefore be useful to neutralize the external surface acidity either by some bulky basic molecules or by terminating the synthesis with some Al free layers of siliceous zeolite. 

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