**3. Simulation of hypothetic binary alloys**

**2. Phase decomposition in alloys**

224 Modeling and Simulation in Engineering Sciences

**Figure 1.** Miscibility gap in a hypothetic A-B phase diagram.

The formation of phases in alloys usually takes place by nucleation mechanism, growth mechanism, or spinodal decomposition mechanism, which is followed by the coarsening of phases in alloy systems. These three processes can be analyzed using the phase-field method, and their results can also be compared with the fundamental theories of phase transformations such as the Cahn-Hilliard spinodal decomposition theory [4] and the Lifshitz-Slyozov-Wagner (LSW) diffusion-controlled coarsening theory [8]. The phase decomposition that takes place by the spinodal decomposition mechanism is distinguished from the phase separation that occurs by nucleation and growth mechanism by the formation of an initial composition modulation, which shows an increase in the modulation amplitude with time. In contrast, the phase formation by nucleation and growth predicts that the formed phase has a composition very close to the equilibrium one from the start to the finish of phase transformation [9]. Besides, the spinodal decomposition is usually associated with the presence of a miscibility gap in the equilibrium phase diagram, as shown in **Figure 1**. The miscibility gap is the equilibrium line and there is only one α phase for compositions and temperatures above this line, whereas a mixture of two phases, α1 and α2, is present inside the miscibility gap. This figure also shows the existence of the chemical spinodal located within the miscibility gap. A supersaturated αsss phase is expected to decompose spinodally into a mixture of A-rich α1 and B-rich α2 phases for an alloy composition after heating at a temperature higher than that of the miscibility gap and then quenching and heating or aging at a temperature lower than the chemical spinodal. The miscibility gap in **Figure 1** is usually related to the plot of free energy versus composition shown in **Figure 2**. This figure shows the free energy curve shape changes as the temperature decreases. This type of curve is known as the spinodal curve, and it indicates that any alloy composition is in unstable state and thus it is expected to decompose into a mixture of A-rich α1 and B-rich α2 phases. The minimum and saddle points at each temperature of the spinodal curve correspond to the equilibrium and chemical spinodal shown in **Figure 1**.

In the numerical simulation of the phase decomposition for a hypothetic A-B binary alloy, the nonlinear Cahn-Hilliard equation [7] can be rewritten as follows:

$$\frac{\partial \mathbf{c}\_{\cdot}(\mathbf{x},t)}{\partial t} = M\_{\cdot} \nabla^{2} \left( \frac{\partial f\_{\circ}(\mathbf{c})}{\partial \mathbf{c}\_{\cdot}} - K\_{\cdot} \nabla^{2} \mathbf{c}\_{\cdot} \right) \tag{7}$$

The local chemical free energy *f*o*(c)* was assumed to follow the following equation [10]:

$$\mathbf{f}\_0 = -(\mathbf{c} - \mathbf{0}.\mathbf{5})^2 + 2.5(\mathbf{c} - \mathbf{0}.\mathbf{5})^4 \tag{8}$$

This equation represents a spinodal curve similar to those shown in **Figure 2**. In the first case of simulation, the mobility *Mi* was considered to be constant and equal to 1, and the compo‐ sition *c* is equal to 0.4. The calculated concentration profiles for different times are shown in **Figure 3**. The initial modulation amplitude increases with time which confirms that the phase decomposition occurs by the spinodal decomposition mechanism [4]. Besides, the initial composition modulation forms a mixture of A-rich α1 and B-rich α2 phases as a result of heating at a temperature with mobility *Mi* equal to 1. An advantage of the phase-field method is that the microstructure evolution can be obtained by plotting the concentration in two dimensions. **Figure 4** shows the calculated microstructures as a function of time. The black and white zones correspond to the A-rich α1 and B-rich α2 phases, respectively. The morphology of the decomposed phases is irregular and interconnected as predicted by the Cahn-Hilliard theory of spinodal decomposition [4]. The coarsening process of the decomposed phases is also observed for the longer times.

**Figure 3.** Concentration profiles for the numerical simulation of *Mi* = 1 and *c* = 0.4.

**Figure 4.** Microstructure evolution for *Mi* = 1 and *c*= 0.4 for (a) 0 h, (b) 0.03 h, (c) 0.6 h, (d)2.8 h, and (e) 6.3 h.

In the second example, the mobility *Mi* was not constant and defined with the following equation:

$$M\_i = \mathbf{l} - \alpha \mathbf{c}^2 \tag{9}$$

where *α* = 1 and the nonlinear Cahn-Hilliar equation was modified as follows:

at a temperature with mobility *Mi* equal to 1. An advantage of the phase-field method is that the microstructure evolution can be obtained by plotting the concentration in two dimensions. **Figure 4** shows the calculated microstructures as a function of time. The black and white zones correspond to the A-rich α1 and B-rich α2 phases, respectively. The morphology of the decomposed phases is irregular and interconnected as predicted by the Cahn-Hilliard theory of spinodal decomposition [4]. The coarsening process of the decomposed phases is also

= 1 and *c* = 0.4.

= 1 and *c*= 0.4 for (a) 0 h, (b) 0.03 h, (c) 0.6 h, (d)2.8 h, and (e) 6.3 h.

<sup>2</sup> 1 *M c <sup>i</sup>* = a

was not constant and defined with the following

(9)

observed for the longer times.

226 Modeling and Simulation in Engineering Sciences

**Figure 3.** Concentration profiles for the numerical simulation of *Mi*

**Figure 4.** Microstructure evolution for *Mi*

equation:

In the second example, the mobility *Mi*

$$\frac{\partial \mathbf{c}\_i(\mathbf{x}, t)}{\partial t} = \nabla \left[ M\_i \nabla \left( \frac{\partial f\_o(c)}{\partial \mathbf{c}\_i} - K\_i \nabla^2 \mathbf{c}\_i \right) \right] \tag{10}$$

The numerical solution of the former partial differential equation conducted to the following concentration profiles (**Figure 5**). The same characteristics described in the previous example are also observed for this case. However, the amplitude of the composition modulation for this case increases faster with time than that for the former case. The microstructure evolution for this case is different from the one shown for the previous case (**Figure 6**). That is, the decom‐ posed phases form a lamellar structure instead of the irregular and interconnected morphology of the previous case. This may be attributed to the variable mobility of the decomposed phases [4].

**Figure 5.** Concentration profiles for the numerical simulation of variable *Mi* and *c* = 0.4.

**Figure 6.** Microstructure evolution for variable *Mi* and *c*= 0.4 for (a) 0 h, (b) 0.03 h, (c) 0.6 h, (d)2.8 h, and (e) 6.3 h.

To complete the numerical simulation of hypothetic binary alloys, the next case includes the presence of an isotropic elastic-strain energy *f*el with a value equal to 1 and *Mi* equal to 1. The nonlinear Cahn-Hilliard equation was modified as follows:

$$\frac{\partial \mathbf{c}\_i(\mathbf{x}, t)}{\partial t} = M\_i \nabla^2 \left( \frac{\partial f\_o(\mathbf{c})}{\partial \mathbf{c}\_i} + \frac{\partial f\_{el}}{\partial \mathbf{c}\_i} - K\_i \nabla^2 \mathbf{c}\_i \right) \tag{11}$$

**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 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.
