P. Bellon

1.15.1 Introduction

Electronic and atomistic processes often dictate the pathways of phase transformations and microstruc­tural evolution in solid materials. For quantitative modeling of these transformations and evolution, it is thus effective, and sometime necessary, to rely on methods using some representation of atoms and of their dynamics, as for instance in molecular dynam­ics simulations (see Chapter 1.09, Molecular Dynamics) and atomistic Monte Carlo simulations (see Chapter 1.14, Kinetic Monte Carlo Simula­tions of Irradiation Effects). While these atomistic methods can now simulate quite accurately the evo­lution of specific alloy systems, these simulations are nevertheless limited to small length scales, from a few to 100 nm. Molecular dynamics is furthermore lim­ited to small time scales, typically in the nanosecond range, although in some cases, new developments have made it possible to obtain atomistic simulations at much longer times (see Chapter 1.14, Kinetic Monte Carlo Simulations of Irradiation Effects).

An alternative modeling approach is to replace the many microscopic degrees of freedom of the system of interest by the few mesoscopic variables that are sufficient to provide a realistic description. This approach has been widely used in many disciplines, and well-known examples are the Fourier and Fick equations, which describe the diffusive transport of heat and chemical species, respectively. This approach is also commonly used in modeling the evolution of point defects, in particular, during irra­diation (see Chapter 1.13, Radiation Damage The­ory and Sizmann1). The work of Cahn and Hilliard2- 5 and Landau and Lifshitz (see for instance Tole — dano and Toledano6) provided a way to include the contributions of interfaces to chemical evolution, thus making it possible to model heterogeneous and multiphase materials. Kinetic models based on these descriptions are broadly referred to as phase field (PF) methods, since the microstructure of a material is fully characterized by a few mesoscopic field variables such as concentration, magnetization, chemical order, or temperature. One key assumption of this approach is that the variables chosen to describe the state of the system vary smoothly across any interface or, in other words, that interfaces are diffuse. This assumption finds a natural justification in the theory of critical phenomena, since the interface thickness diverges at the critical tempera­ture.7 Diffuse interface models offer some advan­tages over sharp interface models,8 in particular, for the modeling of complex microstructures. Fur­thermore, the PF approach can be extended to include macroscopic variables other than the local composition, making it possible to describe chemi­cal order-disorder transitions, solid-liquid reac­tions, displacive transformations, and more recently dislocation glide. PF methods and applica­tions have been recently reviewed by Chen,9 Emmerich,1 and Singer-Loginova and Singer.11 This chapter focuses on solid-solid phase transfor­mations, with a particular emphasis on transforma­tions and microstructural evolution relevant to irradiated materials. While conventional PF model­ing lacks atomic resolution, the main interest in this technique comes from the fact that it can provide the evolution of large systems, exceeding the micrometer scale, over very long time scales, from seconds to centuries. Recent developments have led to the introduction of PF models (PFMs) that pos­sess atomic resolution,12-26 the so-called PF crystal models. This model, which can be seen as a density functional theory for atoms, appears very promising, although at this time it is not clear whether it can reproduce correctly the discrete nature of point — defect jumps from one lattice site to a neighboring lattice site. The PF crystal model is not covered in this chapter, so the interested reader should consult the above references.

This chapter is organized as follows. Section 1.15.2 introduces the key concepts and steps employed in conventional, that is, phenomenological PF modeling, and provides some illustrative examples. Section

1.15.3focuses on important recent developments toward quantitative PF modeling, whereby evolution equations are rigorously derived by coarse-graining a microscopic model. This approach provides a full treatment of fluctuations and thus makes it possible to study fluctuation-controlled reactions, such as nucle — ation of a second phase. The capability of PFMs to reach large time and length scales makes them an attractive tool for simulating the evolution of materi­als relevant to nuclear applications, in particular, for alloys subjected to irradiation. Applying PF modeling to these nonequilibrium materials, however, raises new challenges, as is discussed in Section 1.15.4.1. Some selected results of PF modeling applied to irra­diated materials are presented in Section 1.15.4.2. Finally, conclusions and perspectives are given in Section 1.15.5.