Coherency-Strain Effects

The crystal symmetry changes that accompany order-disorder transitions, discussed in Section 17.1.2, give rise to diffraction phenomena that allow the transitions to be studied quantitatively. In particular, the loss of symmetry is accompanied by the appearance of additional Bragg peaks, called superlattice reflections, and their intensities can be used to measure the evolution of order parameters. 

The driving force for transformation, in Eq. 18.22, was derived from the total Helmholtz free energy, and it was assumed that molar volume is independent 

In crystalline solutions, the developing interfaces are initially coherent—strains are continuous across interfaces. Unless defects such as anticoherency dislocations intervene, the interfaces will remain coherent until a critical stress is attained and the dislocations are nucleated. For small-strain fluctuations, the system can be assumed to remain coherent and the resulting elastic coherency energy can be derived.9 

For example, consider a binary alloy in which the stress-free molar volume is a function of concentration, V(cs). The linear expansion due to the composition change can be inferred from diffraction experiments under stress-free conditions (Vegard s effect) and is characterized by Vegard s parameter, ac [e.g., in cubic or isotropic crystals e ° = e°y 0 = = ac(c — c0)]. The assumption of coherency 

Cahn s early contributions to elastic coherency theory were motivated by his work on spinodal decomposition. His subsequent work with F. Larche created a rigorous thermodynamic foundation for coherency theory and stressed solids in general. A single volume, The Selected Works of John W. Cohn [15], contains papers that provide background and advanced reading for many topics in this textbook. This derivation follows from one in a publication included in that collection [16]. 

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Rhee, W. H., Song, Y. D. and Yoon, D. N., A critical test for the coherency strain effect on liquid film and grain boundary migration in Mo-Ni-(Co-Sn) alloy, Acta MetalL, 35, 57-60, 1987. 

The coherency strains around the GP zones and 0" precipitates generate stresses that heip prevent disiocation movement. The GP zones give the iarger hardening effect (Fig. 10.7). 

A primary focus of our work has been to understand the ferroelectric phase transition in thin epitaxial films of PbTiOs. It is expected that epitaxial strain effects are important in such films because of the large, anisotropic strain associated with the phase transition. Figure 8.3 shows the phase diagram for PbTiOs as a function of epitaxial strain and temperature calculated using Landau-Ginzburg-Devonshire (lgd) theory [9], Here epitaxial strain is defined as the in-plane strain imposed by the substrate, experienced by the cubic (paraelectric) phase of PbTiOs. The dashed line shows that a coherent PbTiOs film on a SrTiOs substrate experiences somewhat more than 1 % compressive epitaxial strain. Such compressive strain favors the ferroelectric PbTiOs phase having the c domain orientation, i.e. with the c (polar) axis normal to the film. From Figure 8.3 one can see that the paraelectric-ferroelectric transition temperature Tc for coherently-strained PbTiOs films on SrTiOs is predicted to be elevated by 260°C above that of... 

Whereas the solubility of Cu in aluminum metal is ca. 5 wt% at temperatures in excess of 500°C, the solubility drops to ca. 0.1 wt% at room temperature. Hence, a metastable alloy is present when the high temperature alloy is rapidly quenched. Subsequent annealing will result in further strengthening similar to what we discussed for martensite. The strengthening effect is thought to occm due to the formation of Cu-rich discs (approx, diameter of 100 atoms, and thickness of ca. 4 atoms) that align themselves preferentially with selected planes of the host A1 lattice, causing coherency strains within the solid-state structure. 

The slab geometry also suffers from other finite-size effects. If the extent of the unit cell parallel to the interface is too small, artificial strain effects axe introduced, because the metal and ceramic axe forced to be coherent by the periodic boundary conditions. Of course, this may be eliminated by enlarging the unit cell, which unfortunately leads to very computerintensive calculations, as is the case with the cluster models. However for the slab model, the oscillations in the electronic density of states are not as dramatic when varying the number of atoms as in the case with clusters. This is because the slab is infinite parallel to the interface. This implies the spectrum is continuous, and the metal slab does not have an artificial band gap, unlike the metal cluster. 

Solid solution hardening of the y matrix resulting from the coherency strains or stiffening effect or larger atom elements such as chromium, molybdenum, and tungsten. 

If the specimen crystal is curved, there will be a range of positions where the diffraction conditions are satisfied even for a plane wave. The rocking curve is broadened. It is simple to reduce the effect of curvature by reducing the collimator aperture. For semiconductor crystals it is good practice never to mn rocking curves with a collimator size above 1 mm, and 0.5 mm is preferable. Curved specimens are common if a mismatched epilayer forms coherently on a substrate, then the substrate will bow to reduce the elastic strain. The effect is geometric and independent of the diffraction geometiy. Table 2.1 illustrates this effect. 

Figure 19.9 Effect of elastic inhomogeneity on elastic strain energy of a coherent...

Clem and Fisher (1958) use a similar treatment as above to derive the solid state nucleation kinetics for new phases at grain boundaries. They neglect orientation of the critical nucleus with respect to the host, strain energy, and coherency effects. Nucleation at the grain boundary interface removes boundary energy. Their treatment yields the following critical values ... 

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