ABSTRACT
Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 8.2.4 Interactions Arising from Elastic Inhomogeneity . . . . . . . . . . . . . . . . . . . . 297 8.2.5 Effect of Fcub + Finh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
8.3 Order-Disorder Phase Transitions and Phase Separation in Binary Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 8.3.1 Ginzburg-Landau Theory of BCCAlloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 8.3.2 Ginzburg-Landau Theory of FCCAlloys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
8.4 Nonlinear Elasticity Model: Incoherent Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 8.4.1 Nonlinear Elastic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 8.4.2 Dislocation Dynamics and Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 8.4.3 Phase Separation Around Dislocations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
In Chapter 7, we discussed the kinetics of phase separation in soft-matter systems. In this chapter, we turn our attention to phase separation in “hard” systems, for example, solids. A variety of spatially modulated domain structures emerge in phase-ordering solids [1,2]. For example, in alloys, a difference arises in the lattice constants of the two phases,which is called the latticemisfit. In structural phase transitions, anisotropically deformed domains of the low-temperature phase emerge in the high-temperature phase. In these cases, elastic strains are induced that radically influence the phase transition behavior. Such strain or elastic effects have long been observed in phase transitions in solids.We need to understand the physical processes involved and make some predictions, which would be of great technological importance. In numerical
studies of phase-ordering phenomena in solids, use has beenmade of a time-dependent Ginzburg-Landau (TDGL) model or a phase-field model, in which the order parameters are coupled to the elastic field in the coarse-grained free-energy functional [1-8]. Such approaches are powerful in understanding the mesoscopic dynamics of domain structures. Some relevant factors in theories and simulations of these processes are (i) elastic anisotropy [1,4], (ii) elastic inhomogeneity (the composition-dependence of the elastic moduli) [2,5,6], and (iii) the simultaneous presence of phase separation and order-disorder phase transitions [1,8].
