**1. Introduction**

In the field of materials science, it is important to analyze different moving free boundary problems in order to understand its effect on the phase transformations that may occur in materials. For instance, the solidification and grain growth, and martensitic transformation are diffusion-controlled and diffusion-less phase transformations, respectively, with this characteristic. One way to overcome this need is the use of the diffuse interface model [1]. Furthermore, the application of the phase-field method to this sort of problem permits the use of the order parameter and phase-field variable which takes into account the composition gradient energy present in a diffuse interface model. For instance [2], the order parameter *φ* could takes values either 0 or 1, which may represent the liquid or solid states, respectively, during the solidification process of a pure metal.

The phase-field method is based on the equations proposed by Cahn-Hilliard [1], Allen-Cahn [2], or Ginzburg-Landau [3]:

$$\frac{\partial \mathcal{C}\_i(\mathbf{x}, t)}{\partial t} = \nabla \left( \sum\_j M\_{ij} \nabla \frac{\partial G\_{s\mu}}{\partial \mathcal{C}\_i(\mathbf{x}, t)} \right) \tag{1}$$

$$\frac{\partial \phi\_i(\mathbf{x}, t)}{\partial t} = -\sum\_{\neq} L\_y \left( \frac{\partial G\_{sys}}{\partial \phi\_i(\mathbf{x}, t)} \right) \tag{2}$$

where *ci (x,t)* and *φ*<sup>ι</sup> *(x,t)* correspond to the field variable, for instance, concentration and order parameter as a function of position *x* and time *t*. *Mij* and *Lij* are the mobility. The free energy of a given system may include, for instance, the following terms [4]:

$$\mathbf{G}\_{\rm syn} = \mathbf{F}\_{\rm c} + \mathbf{F}\_{\rm grad} + \mathbf{F}\_{\rm sr} + \mathbf{F}\_{\rm mag} + \mathbf{F}\_{\rm ele} \tag{3}$$

*Fc* is the local free energy, *F*grad the compositional gradient energy, *F*str the elastic strain energy, and *F*mag and *F*ele the energies corresponding to magnetic and electric effects, respectively.

The composition gradient energy can be defined, for instance, for the field variable, concen‐ tration *c*, by the following mathematical expression [1]:

Application of Phase-Field Method to the Analysis of Phase Decomposition of Alloys http://dx.doi.org/10.5772/64153 223

$$F\_{gvd} = \frac{1}{2} \int\_r k(\nabla c)^2 \, dr \tag{4}$$

where *κ* is the gradient energy coefficient.

to the analysis of phase decomposition during isothermal aging of real binary and ternary alloy systems, such as Fe-Cr, Cu-Ni, and Cu-Ni-Fe. A comparison of simulat‐

In the field of materials science, it is important to analyze different moving free boundary problems in order to understand its effect on the phase transformations that may occur in materials. For instance, the solidification and grain growth, and martensitic transformation are diffusion-controlled and diffusion-less phase transformations, respectively, with this characteristic. One way to overcome this need is the use of the diffuse interface model [1]. Furthermore, the application of the phase-field method to this sort of problem permits the use of the order parameter and phase-field variable which takes into account the composition gradient energy present in a diffuse interface model. For instance [2], the order parameter *φ* could takes values either 0 or 1, which may represent the liquid or solid states, respectively,

The phase-field method is based on the equations proposed by Cahn-Hilliard [1], Allen-Cahn

(,)

*(x,t)* correspond to the field variable, for instance, concentration and order

f

parameter as a function of position *x* and time *t*. *Mij* and *Lij* are the mobility. The free energy

*Fc* is the local free energy, *F*grad the compositional gradient energy, *F*str the elastic strain energy, and *F*mag and *F*ele the energies corresponding to magnetic and electric effects, respectively.

The composition gradient energy can be defined, for instance, for the field variable, concen‐

å (1)

¶ æ ö ¶ = - ç ÷ ¶ ¶è ø <sup>å</sup> (2)

G = sys c grad str mag ele F + F +F +F +F (3)

(,) . (,) *i sys ij j i*

*i sys ij j i*

*x t G L t x t*

*c xt G M t c xt* ¶ æ ¶ö =Ñ Ñ ç ÷ ¶ ¶ è ø

(,)

f

of a given system may include, for instance, the following terms [4]:

tration *c*, by the following mathematical expression [1]:

ed results with experimental ones is also included.

during the solidification process of a pure metal.

[2], or Ginzburg-Landau [3]:

*(x,t)* and *φ*<sup>ι</sup>

where *ci*

**1. Introduction**

222 Modeling and Simulation in Engineering Sciences

**Keywords:** phase field method, phase decomposition, alloys, aging

One of the main advantages of phase-field method is that this method permits to follow the microstructure evolution in two or three dimensions as the time of phase transformations progresses. Thus, the morphology, size, and size distribution could be determined to follow their corresponding growth kinetics. Additionally, the evolution of chemical composition can also be followed during the phase transformations.

To solve either of the partial differential equations, Eqs (1) or (2), several numerical methods have been used such as finite difference method, difference volume method, and finite element method [5]. The use of explicit finite difference method is simple and good alternative to solve this type of differential equations. For instance, the finite difference method can be used to solve a simple partial differential equation such as the simplified one-dimension equation of the second Fick's law:

$$\frac{\partial \mathbf{c}}{\partial t} = D \frac{\partial^z \mathbf{c}}{\partial \mathbf{x}^2} \tag{5}$$

where *D* is the diffusion coefficient. The finite difference solution can be approximated as follows [6]:

$$\frac{\mathbf{c}\_{i}^{l+1} - \mathbf{c}\_{i}^{l}}{\Delta t} = D \frac{\mathbf{c}\_{i+1}^{l} - \Delta \mathbf{c}\_{i}^{l} + \mathbf{c}\_{i-1}^{l}}{\Delta \mathbf{x}^{2}} \tag{6}$$

where *t* indicates the time and *t* + 1 is equal to t + Δ*t* being Δ*t* the time step. The spacing between two nodes is Δ*x*, the distance step. The node number corresponds to *i*. *ci <sup>t</sup>*+1 indicates the calculated concentration for the node *i* in the next time step, *t* + Δ*t* and *ci t* that of the previous time *t*.

This chapter is mainly focused on the application to the phase decomposition by the spinodal decomposition mechanism during the isothermal aging of hypothetical binary alloys using the nonlinear Cahn-Hilliard equation [7]. The effect of main parameters such as the atomic mobility of alloy and elastic-strain energy on the microstructure evolution and growth kinetics is analyzed. To conclude, the application of phase-field method is extended to the analysis of spinodal decomposition during isothermal aging of real binary and ternary alloy systems, such as Cu-Ni, Fe-Cr, and Cu-Ni-Fe. A comparison of simulated results with experimental ones is also included.
