Grain boundary energy defined

A necessary condition for densification to occur is that the grain boundary energy 7gb be less than twice the solid/vapor surface energy This implies that the equilibrium dihedral angle 0 shown in Fig. 10.3 and defined as... 

In materials with high anisotropy in the grain boundary energy and, thus, with faceted grain boundaries, the angles between boundaries may not be uniquely defined by an interfacial tension condition because of torque on facets. ... 

The resulting equilibrium concentrations of these point defects (vacancies and interstitials) are the consequence of a compromise between the ordering interaction energy and the entropy contribution of disorder (point defects, in this case). To be sure, the importance of Frenkel s basic work for the further development of solid state kinetics can hardly be overstated. From here on one knew that, in a crystal, the concentration of irregular structure elements (in thermal equilibrium) is a function of state. Therefore the conductivity of an ionic crystal, for example, which is caused by mobile, point defects, is a well defined physical property. However, contributions to the conductivity due to dislocations, grain boundaries, and other non-equilibrium defects can sometimes be quite significant. 

The role of the grain boundary plane needs to be established, especially in view of anisotropic superconducting order parameters. In thin films, considerably smaller grain boundary junctions than those used in our work would be feasible and it should thus be possible to prepare samples with reasonably well defined facets. Of course, at inclinations such as for the 45°/[001] grain boundaries, where asymmetric facets dominate supposedly because of their much lower energy than other inclinations (see Fig. 10.4), symmetric facets may not form, at least not exclusively. Whenever there is coexistence of... 

Effective medium theory has been used to estimate a wide variety of solid state energies and structures. Examples include relaxations and reconstructions of clean and adsorbate-covered metal surfaces, bulk and surface phonons, interstitial defects, and grain boundaries.Furthermore, because energies are relatively well defined in this approach, it can be used to understand the underlying driving forces leading to defect structures and energies. ... 

The Wulff plot is the conventional plot for surfaces an example of such a plot is shown in Figure 13.1. The most important point is that you have cusps in y versus 0. In ceramics, this certainly occurs if the energy were isotropic, the Wulff plot would be a circle. What is not certain is how many cusps you have. It is instructive to compare surfaces with grain boundaries (section 14.3). In the latter case it is necessary to define the orientation of the grainboundary plane, i.e., having fixed 0, you must still fix n, the normal to both grains. 

Whether a pore shrinks or not is determined by the free energy change that accompanies the change in pore size. If the pore shrinks, there will be a decrease in the free energy due to the decrease in the pore surface area, while there is also an increase due to the increase in the grain-boundary area. If the pore shrinks, the former (decrease) must be larger than the latter (increase). The equilibrium shape of the pore is determined by the dihedral angle if/, which is defined by... 

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