Elastic strain interactions

Precipitate microstructures are important for the strength and hardness of many alloys [196-210]. A number of experimental [211-228] and theoretical [220,229-247] investigations have shown that the development of precipitate morphologies is influenced by elastic interactions (El) resulting from a lattice misfit between matrix and precipitates and from an externally applied elastic strain. 

When packing progressively tetrahedral defects in the fluorite lattice, it is assumed that they interact with the lattice (elastic strains) and with one another (dipolar interaction). Both mechanisms of interaction are strongly dependent on the local charge distribution within the coordination tetrahedron, in which the defect is formed. 

Most of the experimental results on CJTE can be explained on the basis of molecular field theory. This is because the interaction between the electron strain and elastic strain is fairly long-range. Employing simple molecular field theory, expressions have been derived for the order parameter, transverse susceptibility, vibronic states, specific heat, and elastic constants. A detailed discussion of the theory and its applications may be found in the excellent review by Gehring Gehring (1975). In Fig. 4.23 various possible situations of different degrees of complexity that can arise in JT systems are presented. 

Buckingham theory and, 330 chain decomposition, 325 cooperative behavior of defects, 338 deuterium shifts, 361 dispersive interactions, 330 elastic strain, 333 electrostatic coupling, 352 exchange of positions, 351 heteroisotopic pairs, 348... 

During compaction, primary particles are packed, re-arranged and can undergo deformation and possibly breakage. These events can occur sequentially or in parallel. The mechanical strength of a tablet may strongly depend on the mechanical properties of the primary particles and the particle-particle interactions within it. It is essential that the particles deform plastically or rupture since the stored elastic strains can weaken the tablet on release (Roberts and Rowe, 1987). 

In addition to the elastic energy, of course, the chemical interaction energy between metal and ceramic must be accounted for. The possibility of competition between elasticity and chemistry exists such that a rather elastically strained interface combination may be more stable than an unstrained one, because the strained one may have favorable chemical bonding between metal and ceramic atoms at the interface. Therefore, other interface combinations than the least strained need to be considered in general. 

If chemical bonding is considered to be too short range to account for possible interactions between CS planes we must look for other interactions which persist over longer distances in the solid. The two which come to mind are electrical interactions, such as electrostatic forces, and physical interactions, such as elastic strain. Both of these are amenable to theoretical analysis, and in the last two years a number of papers concerning these analyses has appeared in the literature. We will summarize the results so far obtained in the Sections below. 

First, let us consider the origin of the elastic strain. It is easiest to examine the ReOs structure for this purpose. In the fully oxidized ReOj structure each metal atom is surrounded by six oxygen atoms in octahedral co-ordination, as shown in Figure 39 (a). When a CS plane forms, the metal atoms are no longer shielded from each other by an intervening oxygen atom, but are rather more exposed to each other and thus able to interact with each other as shown in Figure 39 (b). The mode of this... 

At the simplest level, calculations of the elastic-strain energy in the matrix between a pair of CS planes, (1/5)2, allows the strain energy in an ordered array of CS planes to be estimated. The earliest way to do this is to add the strain energy in the matrix between each pair of CS planes in the array. This assumes, of course, that there is no interaction from next-nearest neighbour CS planes, and that each CS plane is an effective screen to the forces causing strain in the matrix. The values calculated in this way for the elastic-strain energy between a pair of CS planes is shown in Figure 41. 

Figure 42 The elastic strain energy per unit volume expressed in arbitrary units, for an ordered array of 10 w CS planes in an ReOs type of crystal matrix, plotted as a function of the composition of the crystal x in A/Ox. These values were obtained by considering only elastic strain in the matrix between the CS planes and ignoring next-nearest neighbour interactions (Reproduced by permission from/. Solid State Chem., 1978, 24, 131)...

Electronic Interactions.—As mentioned above, elastic strain is not the only contender for long-range interactions in these materials, and electronic interactions are also likely candidates. The problem here is that it is difficult to know precisely the valence states of the atoms in these structures. In addition, the electrical properties of the materials are not always well known and conductivity data for many of these phases has not yet been obtained. In the literature only electrostatic interactions and polaron interactions have been considered. 

The cooperative Jahn-Teller type interaction is obtained by the orbital-lattice coupled model [12,13]. Let us consider the Hamiltonian with the Jahn-Teller coupling g, the kinetic energy and the lattice potential for the Jahn-Teller phonon mode, the orbital-strain interaction, and the elastic-strain energy. The Jahn-Teller distortion mode Qi around a metal site i is represented by the Jahn-Teller phonon coordinates k- By introduce the canonical transformation for k, the orbital and lattice degrees of freedom are separated. The final form of the interaction between the inter-site orbitals given by... 

Slonczewski JC, Thomas H (1970) Interaction of elastic strain with the structural transition of strontium titanate. Phys Rev B 1 3599-3608... 

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