**4. Simulation in real binary alloys**

To complete the numerical simulation of hypothetic binary alloys, the next case includes the

*i i i i i*

**Figure 7** illustrates the concentration profiles for this case and the same characteristics, observed in the others, are also present. That is, the modulation amplitude increases with time. Besides, a mixture of A-rich α<sup>1</sup> and B-rich α<sup>2</sup> phases is formed after aging. The microstructure evolution is similar to that of the first case for short times. That is, the morphology of the decomposed phases is irregular and interconnected, which is known as percolated structure [4] (**Figure 8**). Nevertheless, one of the decomposed phases takes a cuboid shape for the longest

¶ ¶¶ æ ö = Ñ + -Ñ ç ÷ ¶ ¶¶ è ø (11)

equal to 1. The

presence of an isotropic elastic-strain energy *f*el with a value equal to 1 and *Mi*

2 2 (,) () *<sup>i</sup> <sup>o</sup> el*

*c xt f c f <sup>M</sup> K c t cc*

nonlinear Cahn-Hilliard equation was modified as follows:

228 Modeling and Simulation in Engineering Sciences

times, which is attributed to the isotropic elastic-stain energy [9].

**Figure 7.** Concentration profiles for the numerical simulation of *f*el = 1 and *c* = 0.4.

**Figure 8.** Microstructure evolution for *f*el = 1 and *c* = 0.4 for (a) 0 h, (b) 0.03, (c) 0.6 h, (d)2.8 h, and (e) 6.3 h.

In the next part of this chapter, the numerical simulation of the phased decompositions of Cu-Ni and Fe-Cr alloys after aging at different temperatures for different times is shown. Simu‐ lated results are compared to the experimental ones. To begin, the Cu-Ni alloys are widely used in different industrial applications. The equilibrium phase diagram is shown in **Fig‐ ure 9**. This diagram has a miscibility gap located at temperatures lower than 350°C [11]. Thus, a supersaturated solid solution, formed by heating above 350°C and then quenching, is expected to decompose spinodally into a mixture of Cu-rich and Ni-rich phases after aging at a temperature lower than 350°C. Nevertheless, the growth kinetics of spinodal decomposition is very slow due to the low atomic diffusivity at these temperatures [7]. Thus, the application of the phase-field method to analyze the spinodal decomposition seems to be a good alternative because of the slow kinetics.

**Figure 9.** Equilibrium Cu-Ni phase diagram [11].

The nonlinear Cahn-Hilliard equation, Eq. (7), was solved to analyze the phase decomposition in these alloys. The local energy *f*o was defined using the regular solution model as follows [7]:

$$f\_o = f\_{Cu}\mathcal{c}\_{Cu} + f\_{Ni}\mathcal{c}\_{Ni} + \Omega\_{Cu-Ni}\mathcal{c}\_{Cu}\mathcal{c}\_{Ni} + RT\left(\mathcal{c}\_{Cu}\ln\mathcal{c}\_{Cu} + \mathcal{c}\_{Ni}\ln\mathcal{c}\_{Ni}\right) \tag{12}$$

where *R* is the gas constant, *T* is the absolute temperature. *f*Cu and *f*Ni are the molar free energy of pure Cu and Ni, respectively, and *ΩCu-Ni* is the interaction parameter. The atomic mobility

*Mi* is related to the interdiffusion coefficient *D* − i as follows:

$$
\overline{D}\_l = M\_i \left( \frac{\partial^2 f\_o}{\partial \mathbf{c}\_l^{\;\;\;\;\prime}} \right) \tag{13}
$$

The interdiffusion coefficient *D* − *<sup>i</sup>* was assumed to be defined as follows [4]:

$$
\overline{D}\_l = D\_{\text{v}l} \mathcal{c}\_{\text{Cu}} + (1 - \mathcal{c}\_{\text{Cu}}) D\_{\text{Cu}} \tag{14}
$$

The gradient energy coefficient *K* was determined as proposed by reference [4]:

$$K = \left(\frac{2}{3}\right) h\_{0.5}^M r\_0^2 \tag{15}$$

where *h*0.5 <sup>M</sup> is the mixing heat per unit volume at *c* = 0.5 and *r*<sup>o</sup> is the nearest neighbor distance. The heat of mixing *h*M was determined according to the next equation [4]:

$$h^M = \mathfrak{c}\_{\text{Cu}} \mathfrak{c}\_{N} \mathfrak{Q}\_{\text{Cu}\text{-}N\text{l}} \tag{16}$$

The thermodynamic, diffusion, crystal lattice, and elastic parameters for the microstructure simulation were obtained from the literature [12–15] and these are shown in **Table 1**. The effect of coherency elastic-strain energy was considered to be present during the phase decomposi‐ tion of Cu-Ni alloys in spite of the similar lattice parameters of copper and nickel [13]. This elastic-strain energy was introduced into Eq. (7), according to the simple definition proposed by Hilliard [4]:


**Table 1.** Values of lattice, diffusion, thermodynamic and elastic constants.

$$\int\_{\mathcal{A}} f\_{\mathcal{A}} = A \Big| \eta^2 \, Y \left( \mathcal{C} - \mathcal{c}\_0 \right)^2 d\mathbf{x} \tag{17}$$

where *A* is the cross-sectional area and *Y* is an elastic constant defined by the elastic stiffness constants, *c*11, *c*12, and *c*44, for the Cu-rich and Ni-rich phases. The parameter *η* is equal to dln*a*/ d*c*. In the case of fcc metals, the elastic energy will be a minimum for the <100> crystallographic directions, and thus the *Y* value can be assumed similar to that corresponding to an isotropic material [4]:

Application of Phase-Field Method to the Analysis of Phase Decomposition of Alloys http://dx.doi.org/10.5772/64153 231

$$Y\_{<100>} = c\_{11} + c\_{12} - 2\left(\frac{c\_{12}^2}{c\_{11}}\right) \tag{18}$$

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

*<sup>i</sup>* (1 ) *D Dc c D* = +- *Ni Cu Cu Cu* (14)

è ø (15)

*Cu Ni Cu Ni h cc* = W - (16)

The gradient energy coefficient *K* was determined as proposed by reference [4]:

The heat of mixing *h*M was determined according to the next equation [4]:

*M*

*Ω*Cu-Ni (J mol−1) [12] (8366.0 + 2.802T) + (−4359.6 + 1.812T)(*c*Cu-*c*Ni)

**Parameter Ni-Cu alloys** Crystal lattice parameter *a*(nm) [13] 0.360 *η* (nm) [13] 0.0016

**Table 1.** Values of lattice, diffusion, thermodynamic and elastic constants.

where *h*0.5

230 Modeling and Simulation in Engineering Sciences

by Hilliard [4]:

Diffusion coefficient *D* (cm2

*cij* (J m−3) Cu/Ni [15]

material [4]:

2 3 *<sup>M</sup> K hr* æ ö <sup>=</sup>ç ÷

2 0.5 0

<sup>M</sup> is the mixing heat per unit volume at *c* = 0.5 and *r*<sup>o</sup> is the nearest neighbor distance.

The thermodynamic, diffusion, crystal lattice, and elastic parameters for the microstructure simulation were obtained from the literature [12–15] and these are shown in **Table 1**. The effect of coherency elastic-strain energy was considered to be present during the phase decomposi‐ tion of Cu-Ni alloys in spite of the similar lattice parameters of copper and nickel [13]. This elastic-strain energy was introduced into Eq. (7), according to the simple definition proposed

s−1) [14] Cu 1.5–2.3 exp (−230,000–260,000 J mol−1)/RT

*c*11 = 16.84 × 1010 *c*12 = 12.14 × 1010 *c*44 = 7.54 × 1010

2 2 <sup>0</sup> ( ) *el f A Y c c dx* = h

where *A* is the cross-sectional area and *Y* is an elastic constant defined by the elastic stiffness constants, *c*11, *c*12, and *c*44, for the Cu-rich and Ni-rich phases. The parameter *η* is equal to dln*a*/ d*c*. In the case of fcc metals, the elastic energy will be a minimum for the <100> crystallographic directions, and thus the *Y* value can be assumed similar to that corresponding to an isotropic

Ni 17–35 exp (−270,000–300,000 J mol−1)/RT

*c*11 = 24.65 × 1010 *c*44= 12.47 × 1010

ò (17)

$$\mathbf{c}\_{ij} = \mathbf{c}\_{ij}^{\text{Cu}} \ \mathbf{c}\_{\text{Cu}} + \mathbf{c}\_{y}^{\text{Ni}} \ (\mathbf{l} - \mathbf{c}\_{\text{Cu}}) \tag{19}$$

Considering the elastic strain energy, *f*el, Eq. (7) was rewritten as Eq. (11).

The microstructural simulation was carried out using the finite difference method with 101 × 101 points square grid with a mesh size of 0.25 nm and a time-step size of 10 s. The simulations were performed for the Ni-30at.% Cu alloy at temperatures between 250 and 322°C for different times.

**Figure 10** shows the numerically calculated plots of Cu concentration versus distance for the Ni-30at.%Cu alloy solution treated (0 h) and aged at 300°C for different times. There is an increase in the modulation amplitude with aging time. The increase in amplitude at this temperature confirms that the phase decomposition occurs spinodally in this alloy. The long simulated aging times also confirm that the growth kinetics of phase decomposition is very slow in this alloy system.

**Figure 10.** Ni-30at.%Cu alloy aged at 300°C.

The simulated microstructures of the Ni-30at.%Cu alloy aged at 300°C for 0, 222, 278, 361, and 444 h are shown in **Figure 11 (a–e)**, respectively. The black and white regions correspond to the Cu-rich and Ni-rich phases, respectively. It can be seen that the morphology of the decomposed phases is irregular and interconnected. The volume fraction of the Ni-rich phase increased with aging time.

**Figure 11.** Microstructure evolution of Ni-30at.%Cu alloy aged at 300°C for (a) 0 h, (b) 222 h, (c) 278 h, (d) 361 h, and (e) 444 h.

The experimental Ne-gas field ion microscopy (FIM) images of the Ni-30at.%Cu alloy aged at 300°C for 500 h is shown in **Figure 12**. There is a good agreement between the calculated and experimental morphologies of the decomposed phases.

**Figure 12.** Ne FIM image of the Ni-30at.%Cu alloy aged at 300°C for 500 h.

In the next paragraphs, a second example about the numerical simulation of real alloys is presented. The Fe-Cr alloys are a very important alloy system since this is used as the basis for different industrial alloys, such as the family of stainless steels. The Fe-Cr equilibrium phase diagram [11] also shows a miscibility gap found at temperatures lower than 500°C (**Fig‐ ure 13**). Thus, the phase decomposition of the supersaturated solid solution into a mixture of Cr-rich and Fe-rich phases is also expected as a result of aging at temperatures lower than 500 °C.

Application of Phase-Field Method to the Analysis of Phase Decomposition of Alloys http://dx.doi.org/10.5772/64153 233

**Figure 13.** Equilibrium Fe-Cr phase diagram [11].

**Figure 11.** Microstructure evolution of Ni-30at.%Cu alloy aged at 300°C for (a) 0 h, (b) 222 h, (c) 278 h, (d) 361 h, and (e)

The experimental Ne-gas field ion microscopy (FIM) images of the Ni-30at.%Cu alloy aged at 300°C for 500 h is shown in **Figure 12**. There is a good agreement between the calculated and

In the next paragraphs, a second example about the numerical simulation of real alloys is presented. The Fe-Cr alloys are a very important alloy system since this is used as the basis for different industrial alloys, such as the family of stainless steels. The Fe-Cr equilibrium phase diagram [11] also shows a miscibility gap found at temperatures lower than 500°C (**Fig‐ ure 13**). Thus, the phase decomposition of the supersaturated solid solution into a mixture of Cr-rich and Fe-rich phases is also expected as a result of aging at temperatures lower than 500

experimental morphologies of the decomposed phases.

232 Modeling and Simulation in Engineering Sciences

**Figure 12.** Ne FIM image of the Ni-30at.%Cu alloy aged at 300°C for 500 h.

444 h.

°C.

The phase decomposition simulation was based on a numerical solution of the nonlinear Cahn-Hilliard equation, Eq. (7). The formulation for this case is very similar to that described previously for the numerical simulation of the phase decomposition in Ni-Cu alloys.

The crystal lattice, thermodynamic, diffusion, and elastic constants for the microstructural simulation in Fe-Cr alloys were taken from references [13–16] and these parameters are shown in **Table 2**. The simulation of phase decomposition was pursued using the explicit finite difference method with 101 × 101 and 201 × 201 points square grids with a mesh size of 0.1 and 0.25 nm and a time-step size up to 10 s. The numerical simulation was performed for the Fe-40at.%Cr alloy aged at 470°C for times from 0 to 1000 h. It is important to mention that the initial composition modulation corresponding to the solution-treated sample was calculated using a random number generator.


**Table 2.** Lattice, diffusion, elastic, and thermodynamic parameters.

The plots of Cr concentration versus distance, concentration profiles, for the Fe-40at.%Cr alloy aged at 470°C for different times are shown in **Figure 14**. These concentration profiles indicate clearly that the supersaturated solid solution decomposed spinodally into a mix of Cr-rich and Fe-rich phases since the modulation amplitude increases as the aging time increases.

**Figure 14.** Concentration profiles 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].

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

**Figure 16.** HR-TEM micrograph of Fe-40at.%Cr alloy aged at 470°C for 250 h.
