*2.2. Kinetics of Rafting*

In the heart of rafting is a stress-limited diffusion process, depending on applied stress, σ*A*, lattice misfit, δ, and the difference between the mechanical property of γ and γ' phases, Δ*G* [3,40]. Both applied stress and inherent misfit stress play important roles in the distribution of internal stress in the grains/crystals consisting of γ and γ' phases due to the dependence of chemical potential on local stress state [40]. Under tensile loading, the difference of the chemical potentials between the γ' phase and the γ phase (γ channels) drives the forming atoms of the γ' phase (Al, Ti, Ta, etc.) in horizontal γ channels into vertical γ channels and drives nearly insoluble atoms (Co, Cr, Mo, Re, W, etc.) away from vertical γ channels. This results in the formation of raft structures perpendicular to the tensile loading, as shown in Figure 6 [41,42].

**Figure 6.** Schematic for the migration of solute atoms under tension in <001> direction.

Assuming that the stress-limited diffusion process is the dominant mechanism controlling the rafting, Fan et al. [34] proposed a von-Mises stress-based criterion in determining the rafting direction. Under concurrent action of external stress and misfit stress, the von-Mises stress in different γ channels can be calculated according to the formulations in Table 1. Dislocations can easily accumulate in γ channels under a large stress, leading to the relief of the misfit stress through the migration of the γ/γ' interface and the directional coarsening in the γ channels with maximum von-Mises stress [34,43].

**Table 1.** Calculation of the von-Mises stress in γ channels under uniaxial loading. Here, σ<sup>0</sup> is applied stress in the global coordinate system, σ*<sup>i</sup>* is the magnitude of the misfit stress, and α is a reduction factor. Adapted from [34].


Geometrical characteristics of the microstructures have been used to determine the degree of rafting, such as the width of γ channels [34,44], the dimension of γ' phase [45], and the shape of γ' phase [46]. The images of the microstructures can be captured through scanning electron microscopy (SEM) and transmission electron microscopy (TEM), and the geometrical characteristics of the microstructures can be then analyzed via image-processing algorithms. There are a few algorithms commonly used to analyze the SEM and TEM images, including Connectivity number method [46], AutoCorrelation Method [47,48], Rotational Intercept Method [48], Fourier analysis [9], and Moment invariants method [49].

Fedelich et al. [44] introduced a dimensionless variable ξ for the analysis of rafting as

$$\xi = \frac{w(t) - w\_{\text{cube}}}{w\_{\text{raft}} - w\_{\text{cube}}} \tag{1}$$

where *w*(*t*) is the width of the γ channel at time *t*, *wcube* and *wra f t* are the channel widths of the cubic structure and the raft structure, respectively. Their numerical values are correlated to the volume fraction of γ' phase, *f*γ, and the microstructure periodicity, λ. The dimensionless variable, ξ, varies in a range of 0 (initial cubic morphology) to 1 (complete raft morphology).

Tinga et al. [45] proposed an evolution law for the dimension of γ' phase under the action of a multiaxial stress as

$$\dot{L}\_{i} = \frac{-3}{2} L\_{i} \left[ \frac{\sigma\_{\text{ii}}^{\prime}}{\sigma\_{M} + \sigma\_{\delta}} \right] \frac{A^{\*}}{L\_{0}} exp \left[ \frac{-Q - \sigma\_{M} l L\_{T}}{RT} \right] \text{ ( $i = 1, 2, 3$ )}\tag{2}$$

where *Li* are the dimensions of γ' phase along three orthogonal directions, σ *ii* are the diagonal components of deviatoric stress tensor, σ*<sup>M</sup>* is von-Mises stress, and *UT* is thermal shear-activation volume, *Q*,*R*, and *T* are activation energy, gas constant, and absolute temperature, respectively, *A*<sup>∗</sup> and σδ are constants, and *L*<sup>0</sup> is the dimension of γ' precipitate in cubic shape. Desmorat et al. [50] used the width of γ channel and the dimensions of γ' phase as internal variables in the framework of thermodynamics with elasto-visco-plasticity and Orowan stress (dislocation mechanics). They calculated the Orowan stress τ*or*, which acts as a resistance to the dislocation motion in γ matrices, during the rafting in the following formulation,

$$
\tau\_{\rm or} = a\_{\rm or} \frac{\mu b}{w} \tag{3}
$$

where αor is a material constant ranging from 0.238 to 2.15 [51], and μ and *b* are the shear modulus and the magnitude of Burgers vector, respectively. Their simulation results provided quantitative description of the rafting behavior in nickel-based single crystal superalloys.

There exists the interaction between rafting and dislocation motion. Rafting leads to the accumulation of dislocations in the γ channels, resulting in the relief of the misfit stress through the migration of the γ/γ' interface and the directional coarsening of γ' precipitates. Without mechanical loading, rafting can also prevail at high temperature if the plastic strain in superalloys reaches a threshold value [52,53]. The deformation field (plastic strain) is associated with the presence of dislocations in crystal, even though there is no external loading. The dislocation motion in superalloys at high temperature relieves the misfit stress and promotes the directional coarsening of γ' precipitates through rafting. It is the plastic deformation in the γ matrices (channels) that determines the rafting process and the microstructure evolution.
