Boundary energy

Fig. 2.7. Many metals are made up of /wo phases. This figure shows some of the shapes that they con hove when boundary energies dominate. To keep things simple we hove sectioned the tetrokoidecohedral grains in the way that we did in Fig. 2.6(b). Note that Greek letters ore often used to indicate phases. We hove called the major phase a and the second phase (. But y is the symbol for the energy (or tension) of groin boundaries (y J and interphose interfaces (y ).

We saw in Chapter 2 that, when boundary energies were the dominant factor, we couid easiiy predict the shapes of the grains or phases in a materiai. Isotropic energies gave tetrakaidecahedrai grains and sphericai (or iens-shaped) second-phase particies. 

Now boundary energies can only dominate when the material has been allowed to come quite close to equilibrium. If a structural change is taking place the material will not be close to equilibrium and the mechanism of the transformation will affect the shapes of the phases produced. 

Figure 9.9. Relative boundary energy versus misorienlalion angle for boundaries in copper related by various twist angles about [1 0 0] (after Miura et at. 1990).

For the phase stability analysis we follow the method given by Kanamori and Kakehashi of geometrical inequalities and compute the antiphase boundary energy defined by... 

In Figure 1(b) we have plotted the negative of anti-phase boundary energy e = — ( as a function of band filling for Pd Vi- with x = 0.25, 0.5 and 0.75. The number of zeros is in agreement with the arguments based on moments (there has to be at least four zeros). ... 

Figure 1 (a) The nearest neighbor pair interactions and (b) antiphase boundary energies as functions of energy for Pdj,Vi j, alloys x=0.25, x = 0.5 and x = 0.75 ( from top to bottom). Vertical lines mark the Fermi energy for the three different concentrations. 

Applications to the calculation of static grain boundary energies have been described elsewhere [9,10,14],... 

An area of development is the inclusion of code necessary for free energy perturbation calculations for internal, van der Waals and electrostatic energy terms ( ). Another area of development is the option to use lookup tables for both nonbond energies and for constraint energy terms such as a solvent boundary energy term (16). Other features and options are routinely added as needed to aid in the execution of current projects underway within the NIH. 

Consider the formation of a two-dimensional nucleus and show that the Gibbs energy of a critical cluster is inversely proportional to rj. For this purpose introduce a boundary energy which is proportional to the perimeter of the cluster. 

Figure 6.7 Average grain boundary energy, y b, surface energy of crystals at 0 K, ysS, and surface tension of liquid Al, Ag, Au, Ni and Pt as a function of melting temperature 7W 2/3[8, 11],...

The distribution of the liquid is determined by the interfacial energy between the liquid and the solid matrix relatively to the grain boundary energy. An example is shown in Figure 6.11 (a), where an important characteristic of grain boundaries, the... 

BWG and anti-phase boundary energies. The bond-breaking method first described by Flinn (I960) allows an estimate to be made for the structural component of anti-phase boundary (APB) energies. As good agreement can be obtained... 

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