General method for incorporating CALPHAD free energies of mixing into phase field models: Application to the α-zirconium/δ-hydride system

Highlights

• We present a general method to use CALPHAD-based free energies in phase field models.

• The method removes numerical issues while retaining thermodynamic information.

• We demonstrate the method for the α-zirconium/δ-hydride system.

• Planar interface simulations verify that the method yields highly accurate results.

Abstract

This paper presents a general method of incorporating CALPHAD-based free energies into a phase field model. While CALPHAD-based free energy descriptions provide realistic energetics of alloys, their formulations may pose numerical difficulties for phase field simulations. Specifically, the free energies of mixing for different phases may not necessarily be defined over the same concentration ranges, and their derivatives may exhibit highly nonlinear behavior. The method presented approximates the free energies of mixing using piecewise functions to eliminate the aforementioned characteristics while largely retaining the free energy values and the first and second derivative information, which affect the thermodynamic and kinetic behavior of the system. The method is verified by planar interface simulations of the α-zirconium/δ-hydride system. The phase fractions and compositions obtained from the phase field simulations are compared to the values calculated from the common tangent construction on the original free energies. The results indicate a high degree of accuracy.

Introduction

Quantitative microstructural modeling of phase transformations such as solidification and second-phase precipitation is of major technological and scientific importance for materials development and design. One means of modeling microstructural evolution is the phase field approach, which has been successfully employed to simulate phase transformations such as spinodal decomposition [1], [2], [3], [4], coarsening [5], [6], solidification [7], [8], [9], and thin film growth [3], [10], [11], [12], [13]. Comprehensive descriptions and reviews of phase field modeling are found in Refs. [14], [15], [16], [17], [18]. In a phase field model, a microstructure is described by one or more continuous conserved or nonconserved field variables, termed order parameters. An order parameter is generally denoted as $\psi (\mathbf{r},t)$ and indicates the phase at $\mathbf{r}$, where $\mathbf{r}$ is position and t is time. Each phase is designated by a bulk value (e.g., $\psi =1$ for the α-phase and $\psi =0$ for the β-phase), and the value of ψ changes smoothly between the phases. The position of the interface between the phases is described by an intermediate value (e.g., $\psi =0.5$). Thus, the phase field methodology eliminates the need to track the positions of the interfaces explicitly. The free energy of the system can be described as a functional of the order parameters, and the evolution of the system is driven by the reduction of the free energy [4], [19].

The CALPHAD method is a semi-empirical approach for formulating free energies of mixing using known thermodynamic data and equilibrium phase diagrams [20], [21]. The incorporation of realistic CALPHAD-based free energies into phase field models has significantly increased prediction capabilities of phase field modeling [14], [15], [16]. To date, this approach has been applied to steels [22], [23], [24], [25], superalloys [26], [27], [28], [29], [30], [31], [32], [33], and aluminum alloys [34], [35], [36], among others, with studies examining both solidification [22], [26], [32], [34], [36], [37], [38], [39], [40], [41], [42] and solid-state transformations [23], [24], [25], [26], [27], [28], [29], [30], [31], [33], [35], [43], [44], [45].

While CALPHAD-based free energy descriptions may provide realistic thermodynamic information, the presence of natural logarithm terms in their formulation poses numerical challenges for simulation. For example, Ref. [46] indicated the difficulty in performing phase field simulations at low vacancy concentrations for a TiAlN-vacancy system described by CALPHAD-based free energies. In addition, the free energies of mixing may only be defined over a finite concentration range of solute and their derivatives may exhibit undesirable asymptotic behavior. Furthermore, free energies of mixing for different phases may not be defined over the same concentration ranges (e.g., $G^{\alpha }$ may be defined over an atomic fraction range of (0,0.5), while $G^{\beta }$ may be defined over (0,1)). These attributes may be problematic for numerical simulations because solvers may attempt to compute a free energy for a concentration value outside the range for which it is defined. In many phase field models (e.g., the Wheeler–Boettinger–McFadden (WBM) [47], Kim–Kim–Suzuki (KKS) [48], or Welland–Wolf–Guyer (WWG) [49] models), free energy curves must be defined over a large composition range, even at compositions far from equilibrium. However, the computed mobility may be negative when it is calculated from the experimentally obtained diffusivities and the second derivative of the free energy (e.g., in the spinodal region). To avoid such a situation, the CALPHAD free energy must be modified.

Previously proposed methods of incorporating CALPHAD-based free energies into phase field models include direct coupling to a thermodynamic database [26], [50], [51] or polynomial approximations of the free energies [35], [52], [53], [54], [55]. However, direct coupling may incur substantial computational expense by querying external software at each mesh point for every simulation step, and it may necessitate the use of commercial software, which may require costly licensing fees. Approximating the free energy as a polynomial expression alleviates the aforementioned computational cost, but a simple approximation may cause a loss of thermodynamic information. For example, Ref. [35] found that precipitate growth kinetics and the driving force for nucleation are sensitive to the parameterization of the free energy of the matrix and precipitate phases, respectively, in an Al–Cu system.

The α-zirconium/δ-hydride system is of technological importance, particularly for the fuel assemblies of nuclear power reactors. A CALPHAD-based description of the molar free energies of mixing for this system exists in the literature [56]. However, the free energies exhibit several characteristics that are problematic for phase field modeling, including the presence of natural logarithms arising from the entropic contribution to the free energy. In addition, the solubility limit of hydrogen in α-zirconium is low. Therefore, at low supersaturations a numerical solver may attempt to compute a free energy for a hydrogen atomic fraction less than zero, resulting in undefined behavior due to the natural logarithm terms. Furthermore, the α-phase free energy of mixing is defined over a smaller composition range than that of the δ-phase, so that energy evaluations may occur outside the composition range of the α-phase but still within the composition range of the δ-phase.

In this paper, a simple and efficient method of approximating CALPHAD-based free energies for phase field models is proposed. The approximation method was formulated to retain the original free energy in the composition ranges that are present in the evolving system while alleviating numerical challenges. The approach is incorporated into a phase field model for the α-zirconium/δ-hydride system, which was chosen as a test system because it exhibits the characteristics described previously. Planar interface simulations were performed to verify the model. The equilibrium phase fractions and compositions from the simulations were compared to values obtained from the original CALPHAD free energies via a common tangent construction. In addition, an example of δ-hydride precipitate growth is discussed to demonstrate the flexibility of the method.