Coherent spinodal

Figure 3.16 Solvus and spinodal decomposition fields as a function of elastic strain, (a) Strain-free or chemical solvus (b) strain-free spinodal (c) coherent solvus (d) coherent spinodal. From Ganguly and Saxena (1992). Reprinted with permission of Springer-Verlag, New York.

From this descriptive introduction, it follows that the coherent spinodal decomposition is a continuous transport process occurring in a supersaturated matrix. It is driven by chemical potential gradients. Strain energy and concentration gradient energy have to be adequately included in the component chemical potentials. We expect that the initial stages of decomposition are easier to treat quantitatively than the later ones. The basic result will be the (directional) build-up of periodic variations in concentration [J.W. Cahn (1959), (1961), (1968)]. 

The coherency strain energy introduces an additional barrier to spinodal decomposition, which causes a shift on the temperature-composition phase diagram of the chemical spinodal, defined by d2filom/dc% = 0, to the coherent spinodal, defined by... 

Figure 18.8 Relation between chemical and coherent spinodals.

In crystalline solids, only coherent spinodal decomposition is observed. The process of forming incoherent interfaces involves the generation of anticoherency dislocation structures and is incompatible with the continuous evolution of the phase-separated microstructure characteristic of spinodal decomposition. Systems with elastic misfit may first transform by coherent spinodal decomposition and then, during the later stages of the process, lose coherency through the nucleation and capture of anticoherency interfacial dislocations [18]. 

The (3 dependence of the amplification factor in an elastically isotropic crystal (for which R is independent of the direction of 0) is plotted for a temperature inside the coherent spinodal in Fig. 18.9. For (3 < /3crit> the amplification factor R (3) > 0 and the system is unstable—that is, the composition waves in Eq. 18.44 will grow exponentially. The wavenumber /3max, at which dR f3)/df3 = 0, receives maximum amplification and will dominate the decomposed microstructure. Outside the coherent spinodal, where d2 fhom/dc2B+2a2Y(h) > 0, all wavenumbers will have R(/3) < 0 and the system will be stable with respect to the growth of composition waves. 

Figure 18.9 Amplification factor vs. wavenumber plot for an elastically isotropic crystal at a temperature inside the coherent spinodal where d2 fhom /dc% + 2ct2 Y < 0.

In concentrated Al-Zn alloys, the kinetics of precipitation of the equilibrium 0 phase from a are too rapid to allow the study of spinodal decomposition. An Al-22 at. % Zn alloy, however, has decomposition temperatures low enough to permit spinodal decomposition to be studied. For Al-22 at. % Zn, the chemical spinodal temperature is 536 K and the coherent spinodal temperature is 510 K. The early stages of decomposition are described by the diffusion equation... 

Lop] Atom Probe Field Ion Microscopy (AP-FIM) Cu5oNi46Fe4 and Cu44FegNi4g, atmealed 400-650°C. Coherent spinodal... 

Figure D.2 T-x section of a binary phase diagram A-B with strain-free soivus (binodai) a, chemicai spinodai c, coherent soivus b, and coherent spinodal d. Reprinted with permission from The Mineralogical Society of America.

The exsolution process in either binodal or spinodal decomposition may lead to coherent or incoherent interfaces between the unmixed phases (figure 3.15). The... 

Fet us now consider an initial composition C2. We are within the two spinodal points on the Gibbs free energy coherent curve G decomposition takes place spontaneously throughout the entire phase, with modulated compositional fluctuations whose amphtudes increase as the process advances. We always obtain pigeonite plus diopside, although theoretically the orthopyroxene plus diopside paragenesis, also in this case, corresponds to the minimum energy of the system. 

Figure 12-11. Gibbs energy vs. composition curve explaining metastable and unstable conditions inside and outside the (coherent and incoherent) spinodal during a local composition fluctuation.

If this strain energy of the coherent a-a system is added to the other Gibbs energy contributions, both the total Gibbs energy and the spinodal curve are shifted as shown schematically in Figure 12-10. 

J.W. 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]. 

R.J. Livak and G. Thomas. Loss of coherency in spinodally decomposed Cu-Ni-Fe alloys. Acta Metall, 22(5) 589-599, 1974. 

The precipitate in a Ce-Co-Cu-Fe sample was identified by Leamy and Green (1973) as very small (10 nm), coherent particles of 2-17 phase. Hofer (1970) reported for Sm-Co-Cu alloys a separation into a SmCo5 phase with some Cu in solution and a Co-modified SmCu5 phase. This was said to come about by spinodal decomposition around the composition of SmCo3Cu2 where the coercive force... 

Further experiments on the statistics of cavitation in water inclusions in quartz that approach the spinodal have never been reported and would help to provide a coherent picture of the kinetic stability limit. 

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