*4.2. Ni-Al Binary System*

Nickel-based superalloys consist of multielements, including Ni, Al, Re, Co, etc., while Ni and Al are two major elements determining the microstructure evolution [12]. A Ni-Al binary system has been used in different phase-field models to simulate the γ/γ' microstructure evolution in the nickel-based superalloys.

*C*(*r*, *t*) is defined as the concentration of Al, which is used to distinguish γ' phase (mainly Ni3Al) from γ phase (mainly Ni). For a given concentration, *C*, it is equal to *c*<sup>γ</sup> in the γ phase and *c*γ in the γ' phase, and the region with the Al concentration changing from *c*<sup>γ</sup> to *c*γ corresponds to the γ/γ' interface. Note that the Ni-Al phase diagram is used to determine the numerical values of *c*<sup>γ</sup> and *c*γ at a given temperature [82]. A dimensionless parameter, *C* , is sometimes used instead of the concentration, *C*, to distinguish the γ' phase from the γ phase as [26]

$$\mathbf{C}' = \frac{\mathbf{C} - \mathbf{c}\_{\mathcal{V}}}{\mathbf{c}\_{\mathcal{V}'} - \mathbf{c}\_{\mathcal{V}}} \tag{33}$$

with *C* = 1 representing the γ' phase and *C* = 0 representing the γ phase.

There are four different γ' variants when studying the γ/γ' microstructure evolution. This is because there are antiphase domains between different γ' variants, which affect the directional coarsening of γ' phase [79]. Three field parameters are introduced to characterize four different γ' variants: φ1, φ2, φ<sup>3</sup> = φ0{1, 1, 1}, φ0{1,1,1}, φ0{1,1,1}, φ0{1,1,1} [12]. The concentration field, *C*(*r*, *t*), and the field parameters, φ*<sup>i</sup>* (*i* = 1, 2, 3), are controlled by kinetic equations, which are established on the principles of minimum free energy [12] and/or maximum entropy [83].

The principle of maximum entropy has been mainly used in the solidification analysis of alloys. It is very difficult to calculate the change of entropy during plastic deformation. The principle of minimum free energy is more common in the study of the rafting process. With the principle of minimum free energy, the concentration field, *C*(*r*, *t*), and the field parameters, φ*<sup>i</sup>* (*i* = 1, 2, 3), satisfy the Cahn–Hilliard and Allen–Cahn equations as

$$\frac{\partial \mathbb{C}(r, t)}{\partial t} = \nabla \cdot \left( m\_0 \nabla \frac{\delta F}{\delta \mathbb{C}(r, t)} \right) \tag{34}$$

$$\frac{\partial \phi\_i(\mathbf{r}, t)}{\partial t} = -l\_0 \frac{\delta F}{\delta \phi\_i(\mathbf{r}, t)}, \ (i = 1, 2, 3) \tag{35}$$

The local volume fraction of the γ' phase, *f*(*r*, *t*), is used sometime instead of *C*(*r*, *t*) in Equation (34) [20,84]. Note that the local γ'-volume fraction is equivalent to the dimensionless parameter, *C* . The use of *f*(*r*, *t*) makes it easy to extend the phase-field models for the Ni-Al binary system to multicomponent systems. Here, *m*<sup>0</sup> and *l*<sup>0</sup> are the mobility coefficient and the kinetic coefficient, respectively, and *F* is the total free energy consisting of chemical free energy, *Fch*, and strain energy, *Fel*. The chemical free energy can be expressed by Ginzburg–Landau functional approximation as [12]

$$F^{\rm cl} = \int\_{\mathcal{V}} \left[ f\_{\rm homo} + \kappa\_1 |\nabla \mathbf{C}|^2 + \kappa\_2 \sum\_{i=1,3} \left| \nabla \phi\_i \right|^2 \right] \mathbf{dV} \tag{36}$$

The gradients of the concentration field and field parameters in Equation (36) define the numerical resolution and interface thickness in the simulation [18]. The gradient energy coefficients of κ<sup>1</sup> and κ<sup>2</sup> are related to interfacial energy, and their values are adjusted to ensure that the interface thickness of two-phase microstructure can represent real situation [12,85]. The use of *fhomo* is to distinguish the γ phase from the γ' phase and to assure the stability of four different γ' variants at φ0{1,1,1}, φ0{1,1,1}, φ0{1,1,1}, and φ0{1, 1,1}. The strain energy, *Fel*, is calculated as

$$F^{el} = F^{el} + \frac{1}{2} \int\_{V} \overleftarrow{\lambda} : \vec{\varepsilon}^{el} : \vec{\varepsilon}^{el} dV \tag{37}$$

where <sup>=</sup> λ is local elasticity modulus tensor depending on the concentration field, *C*(*r*, *t*). *F<sup>a</sup>* is a homogeneous term, depending on loading condition. For strain-control condition, *Fa* = 0; for stresscontrol condition, *F<sup>a</sup>* = −*V*σ*<sup>a</sup> ij*ε*ij*. Here <sup>σ</sup>*<sup>a</sup> ij* is the components of applied stress, and ε*ij* is the average strain components [13].

The contribution of the hardening free energy, *Fvp*, or plastic energy, *Fpl*, to the total free energy during plastic deformation was also considered in some works [12,19]. However, the partial derivative of *Fvp* or *Fpl* to *C*(*r*, *t*) in Equation (34) is negligible since most studies have been using the same viscoplastic parameters for both the γ phase and the γ' phase, i.e., the hardening or plastic energy function is homogeneous in two phases and independent of the concentration field, *C*(*r*, *t*).

It is worth mentioning that calculations are performed for small deformation in the phase-field simulation. In this situation, the change of the crystal dimensions is negligible and the total strain rate tensor, ˜ . ε, can be divided into three parts as [26,86]

$$
\vec{\dot{\varepsilon}} = \vec{\dot{\varepsilon}}^{cl} + \vec{\dot{\varepsilon}}^{0} + \vec{\dot{\varepsilon}}^{pl} \tag{38}
$$

Here, ˜ . ε *el* , ˜ . ε 0 , and ˜ . ε *pl* are elastic strain, eigenstrain from the <sup>γ</sup>/γ' lattice misfit <sup>ξ</sup>, and plastic strain rate tensors, respectively. The plastic strain rate tensor, ˜ . ε *pl* , is obtained through the theory of crystal plasticity as [86]

$$\hat{\boldsymbol{\varepsilon}}^{pl} = \sum\_{a=1}^{N} \dot{\boldsymbol{\gamma}}^{(a)} \boldsymbol{m}^{(a)} \otimes \boldsymbol{n}^{(a)} \tag{39}$$

Then the shear strain rate . <sup>γ</sup>(α) is calculated through phenomenological or physics-based constitutive models. However, the use of the theory of crystal plasticity for small deformation indicates that the calculated strains are much less than the strains measured in experiments.

The equilibrium equations must be satisfied all the time during the microstructure evolution. The local stress equilibrium is expressed as [12]

$$\begin{cases} \operatorname{div} \tilde{\boldsymbol{\sigma}} = 0 \\ \boldsymbol{\varpi} = \tilde{\boldsymbol{\sigma}}^{\mathfrak{a}} \end{cases} \in V \tag{40}$$

Minimizing the strain energy function, *Fel*, with respect to the displacement or strain components, *ui* or ε*ij*, under given boundary conditions yields [13]

$$\begin{cases} \frac{\delta F^{l}}{\delta u\_{i}^{l}} = 0, \text{strain} - \text{control condition} \\ \frac{\delta F^{l}}{\delta \overline{\varepsilon}\_{ij}} = \sigma^{a}\_{ij'}, \text{stress} - \text{control condition} \end{cases} \tag{41}$$

The Cahn–Hilliard and Allen–Cahn equations and the equilibrium equations constitute the main framework of the phase-field models, which involve the two-way interaction between the concentration field and the energy function (stresses).
