5.2.1. N-Type/P-Type Rafting

There are reports on the use of the phase-field models in the analysis of the rafting during creep deformation in the frameworks of elastic deformation [14,26] and elasto-plastic deformation [8,12,19,67]. In the framework of elastic deformation, the rafting is determined by the sign of applied stress, σA, the γ/γ' lattice misfit, δ, and the difference of mechanical properties between γ and γ' phases, Δ*G*. The results from the phase-field simulation reveal the formation of N-type and P-type raft structures in consistence with the predictions shown in Figure 3. However, the simulation cannot capture the rafting features associated with plastic deformation [12,26] despite the good replication in the types of the raft structure. For example, Gaubert et al. [12] compared the simulated microstructures in elastic framework with the corresponding ones in elasto-plastic framework. They found raft structures

with straight edges in elastic framework and raft structures with wavy-type edges in elasto-plastic framework. In addition, the plastic deformation increased the rafting rate and promoted the formation of raft structures. Wang et al. [8] presented raft structures and the distribution of plastic strain during creep deformation in elasto-plastic framework. As shown in Figure 10, the raft direction was not strictly perpendicular to the loading direction, and the plastic strain almost appeared in horizontal γ channels and around the γ/γ' interface. They pointed out that further plastic strain that occurred around the γ/γ' interface might result in the instability of raft structures.

**Figure 10.** Microstructure evolution (top row) and the corresponding plastic strain field (bottom row) at different instants at 1253 K under 330 MPa from the phase-field simulation: (**a1**,**a2**) t = 0, (**b1**,**b2**) t = 5 h, (**c1**,**c2**) t = 16.83 h, (**d1**,**d2**) t = 32 h, and (**e1**,**e2**) t = 51.39 h. Dark blue in the top row denotes γ matrix, and other four colors represent four different γ' variants (Reproduced with permission from [8] © 2020 Elsevier).

Zhou et al. [89] pointed out that the kinetics for rafting are not solely determined by the difference of mechanical properties between γ and γ' phases, Δ*G*. They revealed that N-type raft structures can form under tensile loading even for homogeneous γ/γ' microstructures (Δ*G* = 1) if there are dislocations present in horizontal γ channels (normal to the loading direction) at initial state. The time needed to form stable P-type raft structures is less than that to form N-type raft structures, which can be attributed to the difference in the diffusion paths of solute atoms.

According to Equations (34) and (36), the atomic migration in γ channels is controlled by the difference of chemical potentials, where forming elements of the γ' phase (mainly Al in Ni-Al binary system) diffuse from the γ channels with large chemical potential to the matrix with small chemical potential. Under compression in <001> direction, the forming atoms of the γ' phase migrate from two types of vertical γ channels (channel 1 and channel 2, see Table 1) to the horizontal γ channel (channel 3) leading to the formation of P-type raft structure. Under tension in <001> direction, atoms migrate from channel 3 to channels 1 and 2, leading to the formation N-type raft structure.

The phase-field models for Ni-Al binary system have been focused on the diffusion of Al, which is likely not enough for multielement superalloys. Multiple-component phase-field models, which contain the main chemical components of nickel-based superalloys, have been developed in order to systematically investigate the effects of alloy elements on the rafting kinetics [42]. As shown in Figure 6, the results reveal that, under tensile loading, the coalescence of γ' precipitates to form raft structures causes the diffusion of Al, Ti, and Ta from the horizontal γ channels to the vertical γ channels and removes atoms of Co, Cr, and Mo from the vertical γ channels. However, some refractory elements, such as Re and W, likely accumulate near the γ/γ' interface.
