**2. Phase decomposition in alloys**

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**.

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

**Figure 2.** Plot of free energy versus composition for the miscibility gap in **Figure 1**.

The Cahn-Hilliard theory of spinodal decomposition [4] was developed by the modification of the second Fick's law equation in order to allow only the growth of the modulation amplitude in composition with a wavelength larger than a critical value. Furthermore, the nonlinear Cahn-Hilliard equation, used in the simulations of this work, has its origin in this theory.
