**1. Introduction**

Turbine blades are critical structural components in modern aircraft engines and function under extreme service conditions, such as high temperature, high pressure, and high stress level. The performance of turbine blades plays an important role in determining the stability and service life of aircrafts. There is a great need to develop high-performing materials for turbine blades.

Nickel-based single crystal superalloys are currently used in turbine blades due to their excellent mechanical properties at elevated temperature, including high mechanical strength and creep resistance. The excellent mechanical properties of the superalloys are attributed to the two-phase microstructure, i.e., γ' precipitate phase and γ matrix phase, with L12 ordered γ' phase being coherently embedded in γ phase of face-centered cubic structure (Figure 1a). The ordered γ' phase with high volume fraction (about 60–70% in most cases) serves as the strengthening phase in the superalloys and hinders the motion of dislocations during creep deformation [1–3].

Under mechanical loading at high temperature, the morphology of the γ' precipitate evolves from cubic shape to plate-like shape, as shown in Figure 1, which is referred to as directional coarsening or rafting. The rafting is a complex process, which can contribute to the hardening of the materials, since the raft structure inhibits dislocation climb around the γ' phase and restricts the motion of dislocations in γ channels [2–4]. However, there are reports that the rafting behavior can also lead to the softening of materials [5,6]. The onset of rafting widens the width of some γ channels which reduces the blocking effect of Orowan stress and causes the bowing of dislocations in the γ channels (Figure 2). This behavior results in the increase of the plastic strain in the corresponding γ channels and the nearby γ/γ' interfaces. Note that the plastic deformation around the γ/γ' interfaces plays a

key role in the rafting process. Therefore, the rafting can lead to the strengthening and softening of nickel-based superalloys during creep, which necessitates the understanding of the evolution of the raft structure under mechanical loading at elevated temperature.

**Figure 1.** SEM images of the typical microstructure of nickel-based single crystal superalloy (DD5): (**a**) initial microstructure without deformation, and (**b**) raft structure after the creep deformation at 980 ◦C.

**Figure 2.** Dislocations bowing through γ channels after long-term aging at 1050 ◦C for 100 h: (**a**) TEM image, and (**b**) schematic of dislocation bowing. In the field of view, some dislocations are present in the γ' phase, which is attributed to the dislocation climb [7] (Reproduced with permission from [8] © 2020 Elsevier).

There are a variety of methods available to investigate the rafting behavior, which can be divided into two categories: experimental methods [1,9] and numerical simulation [10–13]. In general, it is very costly and impractical to use conventional experimental methods to study the rafting behavior due to the need to "record" the changes of the microstructures during creep deformation. Numerical methods, such as finite element method [10], Monte Carlo method [11], and phase-field method [12], have become a major approach to understand the microstructure evolution associated with the rafting process. Among the numerical methods, the phase-field method has become an important technique to analyze the microstructure evolution, which involves phase transition at mesoscale for a long-time period without explicit interface tracking [14–20], and has been used efficiently to simulate the rafting process. Realizing the presence of plastic deformation during the rafting, the theory of crystal plasticity is usually incorporated in the phase-field method to investigate the plastic deformation concurrently presented during the rafting.

In this review, we briefly introduce the rafting behavior in nickel-based single crystal superalloys, together with the crystal plasticity theory and the phase-field method. The models, which are based on the crystal plasticity theory and the phase-field method, respectively, for the analyses of plastic deformation and microstructure evolution are summarized. The applications of the crystal plasticity theory and the phase-field method in studying the rafting behavior are then discussed. Finally, a brief summary is given.
