Influence of the phase diagram on the diffuse interface thickness and on the microstructure formation in a phase-field model for binary alloy

Abstract

A phase-field model is used to investigate the responses of planar interfaces and of eutectic microstructures on the different shapes of the phase diagram in binary alloy systems. Numerical solutions of the dynamic field equations show that the interfacial profile and the thickness of the diffuse boundary layer depend on the segregation of the alloy components. The simulations are presented for different openings of binary phase diagrams. A strong influence of the phase diagram on the evolution of eutectic microstructures is found and quantitatively evaluated by defining an appropriate measure. The results are interpreted in terms of weighting the different contributions in the phase-field equation.

Introduction

In phase-field models (see reviews [1], [2]), the thermodynamic state of regions in a system is described by a continuous order parameter, the so-called phase field $\varphi (x)$. Considering solidification as an example of a phase transition process, the bulk solid and liquid phases can be represented by the values $\varphi =1$ and $\varphi =0$, respectively. The diffuse solid–liquid interface is determined by a transitional region in which the phase field $\varphi (x)$ varies continuously between the values of the bulk phases. For a planar interface in a pure substance under thermodynamic equilibrium conditions, the dependence of the phase-field profile on the spatial coordinate $x$ has the well known form

$${\varphi }_0(x)=\frac{1}{2}\left(1-\tanh \frac{3x}{2\varepsilon }\right)\text{,}$$

where $\varepsilon$ is related to the width of the diffuse interface layer as described in the following. Due to the smooth transition of the order parameter from $\varphi =1$ to $\varphi =0$, we introduce conventional boundaries $x_S$ of the solid ($-\infty <x<x_S$) and $x_L$ of the liquid ($x_L<x<+\infty$) corresponding to the values of the phase field ${\varphi }_S=\varphi (x_S)={\varphi }_0(-\varepsilon )\approx 0.953$ and ${\varphi }_L=\varphi (x_L)={\varphi }_0(+\varepsilon )\approx 0.047$. The thickness $\delta =|x_L-x_S|$ of the phase-field profile ${\varphi }_0(x)$ in Eq. (1) for an interface at equilibrium is equal to $2\varepsilon$ and increases linearly with the parameter $\varepsilon$.

The profile ${\varphi }_0(x)$ is the stationary solution of a nonlinear partial differential equation describing the evolution and dynamics of the solid–liquid interface

$$\frac{2\varepsilon \gamma }{\nu }\frac{\partial \varphi }{\partial t}=2\varepsilon \gamma \frac{{\partial }^2\varphi }{\partial x^2}-\frac{9\gamma }{\varepsilon }{g}_{\text{,}\varphi }-\frac{1}{T}{f}_{\text{,}\varphi }\text{,}$$

for the particular case of a planar interface at thermodynamic equilibrium, i.e. $\partial \varphi /\partial t=0$ and $f_{\text{,}\varphi }=0$. According to phase-field approach in [4], [8], Eq. (2) can be derived from an entropy density functional in a thermodynamically consistent way. The notation ${(\cdot )}_{\text{,}\varphi }$ abbreviates the partial derivative of the functions $g$ and $f$ with respect to $\varphi$. The function $f$ in Eq. (2) is a superposition of free energy densities of the bulk phases, $g(\varphi )={\varphi }^2{(1-\varphi )}^2$ can be chosen of the form of a typical double-well potential, $T$ is the thermodynamic temperature, $\gamma$ is the interface entropy density, and $\nu$ determines the interface mobility for deviations from thermodynamic equilibrium.

The parameter $\varepsilon$ is usually associated with the thickness of the transitional layer between the phases as described by Eq. (1) and plays an important role in numerical simulations. From a computational point of view, one tries to keep $\varepsilon$ (having in mind the interface thickness) as large as possible in order to cover larger microstructure during the simulation. However, a very thick interface leads to a number of undesirable effects. In order to resolve this contradiction, a set of simulations with decreasing $\varepsilon$ can be conducted to obtain convergence of the simulation results in $\varepsilon$ and to iteratively determine an appropriate choice for a suitable value.

In the general case of an alloy system, Eq. (2) contains a driving force $f_{\text{,}\varphi }\ne 0$ related to the bulk free energy densities of the specific phase diagram. This term determines a nonlinear coupling between the phase-field profile and the spatial distributions of other physical fields in the system (e.g. temperature, concentrations, etc.). It has been reported in [6] that the finite interface thickness introduces chemical potential gradients within the interfacial region leading to an effect on the interface velocity and on the relationship between material properties and the coefficients in the phase-field gradient energy. As further shown in [7], a reduction of the interface diffuseness by localizing the solute redistribution into a narrow region of the phase-field profile effectively suppresses anomalous interfacial effects and allows computations of dendritic solidification with quantitatively the same results as the antitrapping model in [5].

In Section 2, we first consider a planar interface of a binary two-phase alloy at equilibrium and analyze the dependence of the interface thickness on the segregation of the alloy components at the interface (i.e. on the term $1/T{f}_{\text{,}\varphi }$ in Eq. (2)). In Section 3, we extend our investigations to 2D numerical simulations of binary eutectic systems and discuss the effect of the phase diagram and of the interface thickness parameter $\varepsilon$ on the microstructure and on the changes in temporal evolution of the volume fractions.