**5. Simulation in real ternary alloys**

**Figure 14.** Concentration profiles of Fe-40at.%Cr alloy aged at 470°C.

234 Modeling and Simulation in Engineering Sciences

**Figure 15.** Microstructure evolution of Fe-40at.%Cr alloy aged at 470°C.

**Figure 15** shows the simulated microstructural evolution of the phase decomposition in the Fe-40-at.%Cr alloy aged at 470°C for times from 10 to 750 h. The white and gray zones represent the Fe-rich and Cr-rich phases, respectively. It can be observed and irregular and intercon‐ nected morphology of the decomposed phases in the alloy aged for times up to 10 h. This morphological characteristic is known as percolated structure, and it has been commonly observed to occur during the early stages of aging in the spinodally decomposed alloys. The HR-TEM micrographs of this alloy aged at 470°C for 250 h shows clearly the presence of spheres corresponding to the Cr-rich phases imbedded in the ferrite phase matrix (**Figure 16**). The decomposed phases present a coherent interface. This shape of decomposed Cr-rich phase is

in good agreement with the simulated microstructure (**Figure 15(e)** and **(f)**) [17].

The numerical simulation of phase decomposition of ternary alloys can be also conducted using the phase-field method. The equilibrium Cu-Ni-Fe phase diagram [18] is shown in **Figure 17** at different temperatures. The presence of a miscibility gap is evident and thus the phase decomposition can also be simulated by the phase-field method.

**Figure 17.** Equilibrium Cu-Ni-Fe phase diagram [16].

The Cahn-Hilliard nonlinear equation for a multicomponent system with a constant mobility can be used for the present simulation, Eq. (7).

The local free energy *f*0 was defined using the regular solution model as follows [19]:

$$f\_o = f\_{Cu}\mathcal{c}\_{Cu} + f\_{Ni}\mathcal{c}\_{Ni} + f\_{Fe}\mathcal{c}\_{Fe} + \Omega\_{Cu-Ni}\mathcal{c}\_{Cu}\mathcal{c}\_{Ni} + \Omega\_{Cu-Fe}\mathcal{c}\_{Cu}\mathcal{c}\_{Fe} + \Omega\_{Ni-Fe}\mathcal{c}\_{Ni}\mathcal{c}\_{Fe} + \dots$$

$$\Omega\_{\rm Cu-Ni-Fe} \mathcal{c}\_{\rm Cu} \mathcal{c}\_{\rm Ni} \mathcal{c}\_{\rm Fe} + RT \left( \mathcal{c}\_{\rm Cu} \ln \mathcal{c}\_{\rm Cu} + \mathcal{c}\_{\rm Ni} \ln \mathcal{c}\_{\rm Ni} + \mathcal{c}\_{\rm Fe} \ln \mathcal{c}\_{\rm Fe} \right) \tag{20}$$

where *R* is the gas constant, *T* is the absolute temperature, *f*Cu, *f*Ni, and *f*Fe correspond to the molar free energy of pure Cu, Ni, and Fe, respectively, and *Ω*Cu-Ni, *Ω*Cu-Fe, *Ω*Ni-Fe, and *Ω*Cu-Ni-Fe represent the interaction parameters. All these thermodynamic constants are shown in **Table 3**.


**Table 3.** Values of lattice, diffusion, thermodynamic, and elastic parameters.

The atomic mobility *Mi* is related to the interdiffusion coefficient *D* i as follows:

$$\overline{D} = M\_i \left(\frac{\partial^2 f\_0}{\partial \mathbf{c}\_i^2}\right) \tag{21}$$

−

The atomic mobility *Mi* was determined using Eq. (21) and the procedure proposed by Honjo and Saito [19]:

$$M\_{\mathcal{N}} = \frac{D\_{\mathcal{N}}}{2\Omega\_{\text{Cu-N}} + 4RT} \tag{22}$$

$$M\_{Fe} = \frac{D\_{Fe}}{2\Omega\_{Cu-Fe} + 4RT} \tag{23}$$

where *D*Ni corresponds to the diffusion coefficient in Cu-50at.%Ni alloy and *D*Fe to the diffusion coefficient of Fe in Cu. The values of *D*Ni and *D*Fe are also indicated in **Table 3**.

The gradient energy coefficient *Ki* was determined as proposed by Hilliard [4]. It was shown in Eq. (15).

Thus, the gradient energy coefficient *Ki* was defined as follows:

( ln ln ln ) *Cu Ni Fe Cu Ni Fe Cu Cu Ni Ni Fe Fe c c c RT c c c c c c* W + ++ - - (20)

(9534.49+2.839T) + (−424.255−0.629T)(*c*Cu-*c*Ni) (48206.0 − 8.446T) + (−5918.0 + 5.017T)(*c*Cu-*c*Fe) (−18298.8 + 5.149T) + (14313.6 − 7.659T)(*c*Ni-*c*Fe)

−7023.9

−532.3

−4154.5

−

æ ö ¶ <sup>=</sup> ç ÷ ¶ è ø (21)



c11= 24.65 × 1010 c12= 14.73 × 1010 c44 = 12.47 × 1010

i as follows:

where *R* is the gas constant, *T* is the absolute temperature, *f*Cu, *f*Ni, and *f*Fe correspond to the molar free energy of pure Cu, Ni, and Fe, respectively, and *Ω*Cu-Ni, *Ω*Cu-Fe, *Ω*Ni-Fe, and *Ω*Cu-Ni-Fe represent the interaction parameters. All these thermodynamic constants are shown in **Table 3**.

−35982.0 − 12.0T

s−1) [14] *D*Ni = 17.0 exp (−279,350 J mol−1)/RT

*c*11 = 16.84 × 1010 *c*12= 12.14×1010 c44= 7.54×1010

is related to the interdiffusion coefficient *D*

2 0

The atomic mobility *Mi* was determined using Eq. (21) and the procedure proposed by Honjo

2 4 *Ni*

2 4 *Fe*

*Cu Fe*

*Cu Ni*

*Ni*

*Fe*

*<sup>D</sup> <sup>M</sup>*

*<sup>D</sup> <sup>M</sup>*

<sup>2</sup> *<sup>i</sup> <sup>i</sup> i*

*<sup>f</sup> D M <sup>c</sup>*

−8.65T−22.64TlnT−3.13×10−3 T2

93.23T−12.54TlnT+1.23×10−3 T2

39.0T−26.61TlnT+1.23×10−3 T2

*D*Fe = 6.1 exp (−266,000 J mol−1)/RT

**Parameter Cu-Ni-Fe alloys**

236 Modeling and Simulation in Engineering Sciences

Crystal lattice parameter *a* (nm) [13] 0.360 *η* (nm) [13] 0.0016

**Table 3.** Values of lattice, diffusion, thermodynamic, and elastic parameters.

*Ω*Cu-Ni *Ω*Cu-Fe *Ω*Ni-Fe

*f*Cu *f*Ni

*Ω*Cu-Ni-Fe (J mol−1) [12]

Diffusion coefficient *D* (cm2

The atomic mobility *Mi*

and Saito [19]:

*f*Fe(J mol−1) [12]

*c*ij (J m−3) Cu/Ni [15]

$$K\_{N} = \frac{1}{12}a^{2}\,\Omega\_{Cu-Ni} \tag{24}$$

$$K\_{F\varepsilon} = \frac{1}{12}a^2 \,\Omega\_{Cu-Fa} \tag{25}$$

where *a* represents the lattice parameter also given in **Table 3**.

The effect of coherency elastic-strain energy was introduced into Eq. (7), according to the simple definition proposed by Hilliard [4]. It was shown in Eq. (17).

The elastic constant *Y* was assumed to be determined with the following equation [4]:

$$Y = \frac{1}{2}c\_{11} + 2c\_{12} \left( 3 - \frac{c\_{11} + 2c\_{12}}{c\_{11} + 2c\_{12}(2c\_{44} - c\_{11} + c\_{12})(l^2 \ m^2 + m^2 \ m^2 + l^2 \ m^2)} \right) \tag{26}$$

where *l*, *m*, and *n* are the direction cosines.

The elastic constants, *cij* were calculated as follows:

$$\mathbf{c}\_{ij} = \mathbf{c}\_{ij}^{\text{Cu} - rich} \ \mathbf{c}\_{\text{Cu}} + \mathbf{c}\_{ij}^{\text{Ni} - rich} \ (\mathbf{l} - \mathbf{c}\_{\text{Cu}}) \tag{27}$$

Considering the elastic strain energy, *f*el, Eq. (7) can be rewritten for the *c*Ni and *c*Fe as follows:

$$\frac{\partial \mathcal{C}\_{\rm{NL}}(\mathbf{x}, t)}{\partial t} = M\_{\rm{NL}} \nabla^2 \left( \frac{\partial f\_o(\mathbf{c})}{\partial \mathbf{c}\_{\rm{NL}}} + \frac{\partial f\_{ol}}{\partial \mathbf{c}\_{\rm{NL}}} - K\_{\rm{NL}} \nabla^2 \mathbf{c}\_{\rm{NL}} \right) \tag{28}$$

$$\frac{\partial \mathcal{L}\_{F\varepsilon}(\mathbf{x}, t)}{\partial t} = M\_{F\varepsilon} \nabla^2 \left( \frac{\partial f\_o(\mathbf{c})}{\partial \mathbf{c}\_{F\varepsilon}} + \frac{\partial f\_{el}}{\partial \mathbf{c}\_{F\varepsilon}} - K\_{F\varepsilon} \nabla^2 \mathbf{c}\_{F\varepsilon} \right) \tag{29}$$

Equations (28) and (29) were solved numerically using the explicit finite difference method with 101 × 101 points square grid with a mesh size of 0.36 nm and a time-step size up to 10 s. The simulations were performed for the Cu-46at.%Ni-4at.%Fe alloy at temperature of 400°C for times from 0 to 200 h. This composition was selected for comparison of the morphology and kinetics of the phase decomposition. It is important to mention that the initial composition modulation corresponding to the solution treated sample was calculated using a randomnumber generator as proposed in reference [19].

**Figure 18** shows the calculated concentration profiles of the Cu-46at.%Ni-4at.%Fe alloy aged at 400°C for times from 0 to 200 h. An increase in the amplitude of the composition modulation with aging time can be noticed in both cases. This fact has been associated with the phase decomposition via the spinodal decomposition mechanism [4]. This behavior is also in good agreement with the experimental evidence reported in the literature [18] for the aging of Cu-Ni-Fe alloys. The calculated Cu and Fe concentration profiles are shown in **Figure 19** for the alloys aged at 400°C for 200 h. These plots indicate clearly that the supersaturated solid solution decomposes into two phases: a Cu-Ni-Fe-rich phase with a poor content of Fe and a Ni-Cu-Fe-rich phase with a lower content of Cu and a higher content of Fe. The decomposed Cu-Ni-Fe and Ni-Cu-Fe phases are in agreement with the miscibility gap of the equilibrium Cu-Ni-Fe phase diagram [17].

**Figure 18.** Cu concentration profile of the Cu-46at.%Cu-4at.%Fe alloy aged at 400°C.

**Figure 19.** Cu and Fe concentration profile of the Cu-46at.%Cu-4at.%Fe alloy aged at 400°C for 200 h.

The simulated microstructural evolution for the Cu-46at.%Ni-4at.%Fe alloy aged at 400°C for different times is shown in **Figure 20**. The white and black zones correspond to the Ni-Cu-Ferich and Cu-Ni-Fe-rich phases, respectively. The morphology is irregular and interconnected, and it has no preferential alignment in a specific crystallographic direction (**Figure 20(a–c)**). This type of microstructure has been named isotropic [4]. The volume fraction of phases is similar because of the small difference in chemical composition. For higher temperatures, the isotropic morphology at the early stages of aging, as the aging progresses, changes to a cuboid or plate shape, crystallographically aligned along the <100> directions because of the low coherency-strain energy associated with these directions [9].

for times from 0 to 200 h. This composition was selected for comparison of the morphology and kinetics of the phase decomposition. It is important to mention that the initial composition modulation corresponding to the solution treated sample was calculated using a random-

**Figure 18** shows the calculated concentration profiles of the Cu-46at.%Ni-4at.%Fe alloy aged at 400°C for times from 0 to 200 h. An increase in the amplitude of the composition modulation with aging time can be noticed in both cases. This fact has been associated with the phase decomposition via the spinodal decomposition mechanism [4]. This behavior is also in good agreement with the experimental evidence reported in the literature [18] for the aging of Cu-Ni-Fe alloys. The calculated Cu and Fe concentration profiles are shown in **Figure 19** for the alloys aged at 400°C for 200 h. These plots indicate clearly that the supersaturated solid solution decomposes into two phases: a Cu-Ni-Fe-rich phase with a poor content of Fe and a Ni-Cu-Fe-rich phase with a lower content of Cu and a higher content of Fe. The decomposed Cu-Ni-Fe and Ni-Cu-Fe phases are in agreement with the miscibility gap of the equilibrium Cu-Ni-

number generator as proposed in reference [19].

238 Modeling and Simulation in Engineering Sciences

**Figure 18.** Cu concentration profile of the Cu-46at.%Cu-4at.%Fe alloy aged at 400°C.

**Figure 19.** Cu and Fe concentration profile of the Cu-46at.%Cu-4at.%Fe alloy aged at 400°C for 200 h.

The simulated microstructural evolution for the Cu-46at.%Ni-4at.%Fe alloy aged at 400°C for different times is shown in **Figure 20**. The white and black zones correspond to the Ni-Cu-Fe-

Fe phase diagram [17].

**Figure 20.** Microstructure evolution of the Cu-46at.%Cu-4at.%Fe alloy aged at 400°C for (a) 50 h, (b) 100 h, and (c) 200 h.

**Figure 21** shows the FIM microstructural evolution of the Cu-46at.%Ni-4at.%Fe alloy aged at 400°C for 50 h. FIM image of the solution treated and quenched sample shows the characteristic concentric ring of the (001) plane in both alloys. In the case of the aged samples, brightly imaged zones correspond to the Ni-Cu-Fe-rich phase and dark zones to the Cu-Ni-Fe-rich phase (matrix). The morphology of the decomposed phases was also cuboids or plates aligned in the <100> directions as the aging progressed.

**Figure 21.** FIM image of the Cu-46at.%Cu-4at.%Fe alloy aged at 400°C for 50 h.
