General method for incorporating CALPHAD free energies of mixing into phase field models: Application to the α-zirconium/δ-hydride system
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 and indicates the phase at , where is position and t is time. Each phase is designated by a bulk value (e.g., for the α-phase and 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., ). 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., may be defined over an atomic fraction range of (0,0.5), while 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.
Section snippets
Methods
To incorporate a CALPHAD-based free energy functional in a phase field model, an implementation must effectively handle any instances where the free energies of mixing for the phases become undefined or discontinuous. These instances may be treated either by numerical exceptions or by modifying the functional such that it yields numerically acceptable behavior while retaining essential thermochemical information. In this paper, we take the latter approach. Below, we first describe a general
Results and discussion
In this section, the equilibrium phase fractions and compositions from the simulations were compared to values obtained from and via a common tangent construction, denoted with PF and CP, respectively. An example of the initial and final conditions of the planar interface simulations is shown in Fig. 3 for at 550 K. An initial hydride plate is present in a supersaturated zirconium matrix; at equilibrium it has grown to its final phase fraction () and the matrix and
Conclusions
The proposed free energy approximation method is a simple and efficient technique to overcome the numerical challenges encountered during phase field simulations that incorporate CALPHAD-based free energies of mixing. The free energies are transformed into piecewise functions composed of three subfunctions that are defined on subdomains of solute composition. Low and high subdomain boundary compositions are chosen within the composition domain over which each free energy is defined. The
Acknowledgments
This research was supported by the Consortium for Advanced Simulation of Light Water Reactors (www.casl.gov) an Energy Innovation Hub (http://www.energy.gov/hubs) for Modeling and Simulation of Nuclear Reactors under U.S. Department of Energy Contract no. DE-AC05-00OR22725. The simulations were performed using the high performance computation resources Fission and Quark at Idaho National Laboratory (INL). Many thanks to the MOOSE team at INL for the continuing, generous help and support with
References (76)
On spinodal decomposition
Acta Metall.
(1961)- et al.
Simulation of convection and ripening in a binary alloy mush using the phase-field method
Acta Mater.
(1999) - et al.
Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method
Acta Metall. Mater.
(1995) - et al.
Extending phase field models of solidification to polycrystalline materials
Acta Mater.
(2003) - et al.
Phase field microelasticity modeling of surface instability of heteroepitaxial thin films
Acta Mater.
(2004) - et al.
An introduction to phase-field modeling of microstructure evolution
CALPHAD
(2008) - et al.
Phase-field modeling of multi-component systems
Curr. Opin. Solid State Mater. Sci.
(2011) - et al.
A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening
Acta Metall.
(1979) - et al.
A phase field model for isothermal solidification of multicomponent alloys
Acta Mater.
(2001) - et al.
A phase field study for ferrite–austenite transitions under para-equilibrium
Scr. Mater.
(2001)
On the formation of widmanstatten ferrite in binary Fe–C—phase-field approach
Acta Mater.
Coupling of multicomponent thermodynamic databases to a phase field modelapplication to solidification and solid state transformations of superalloys
Scr. Mater.
Linking phase-field model to CALPHADapplication to precipitate shape evolution in Ni-base alloys
Scr. Mater.
Three-dimensional phase-field simulations of coarsening kinetics of particles in binary Ni–Al alloys
Acta Mater.
Simulating interdiffusion microstructures in Ni–Al–Cr diffusion couplesa phase field approach coupled with CALPHAD database
Scr. Mater.
Modeling the microstructural evolution of Ni-base superalloys by phase field method combined with CALPHAD and CVM
Comput. Mater. Sci.
Phase-field simulation with the CALPHAD method for the microstructure evolution of multi-component Ni-base superalloys
Intermetallics
Phase-field modelling of as-cast microstructure evolution in nickel-based superalloys
Acta Mater.
A ternary phase-field model incorporating commercial CALPHAD software and its application to precipitation in superalloys
Acta Mater.
Phase field simulation of equiaxed solidification in technical alloys
Acta Mater.
Thermodynamic description and growth kinetics of stoichiometric precipitates in the phase-field approach
CALPHAD
Phase-field model for solidification of ternary alloys coupled with thermodynamic database
Scr. Mater.
A phase-field model coupled with a thermodynamic database
Acta Mater.
A phase-field model for the solidification of multicomponent and multiphase alloys
J. Cryst. Growth
Thermodynamics of grain boundary premelting in alloys. I. Phase-field modeling
Acta Mater.
Phase field modeling of the crystallization of FeOx–SiO2 melts in contact with an oxygen-containing atmosphere
Chem. Geol.
Phase-field modeling of coring during solidification of Au–Ni alloy using quaternions and CALPHAD input
Acta Mater.
Quantitative phase field modeling of diffusion-controlled precipitate growth and dissolution in Ti–Al–V
Scr. Mater.
Phase-field simulations of α to γ precipitations and transition to massive transformation in the Ti–Al alloy
Acta Mater.
Phase-field simulations of intermetallic compound evolution in Cu/Sn solder joints under electromigration
Acta Mater.
Phase-field modelling of spinodal decomposition in TiAlN including the effect of metal vacancies
Scr. Mater.
A generalized computational interface for combined thermodynamic and kinetic modeling
CALPHAD
Incorporating the CALPHAD sublattice approach of ordering into the phase-field model with finite interface dissipation
Acta Mater.
Theoretical and numerical study of lamellar eutectoid growth influenced by volume diffusion
Acta Mater.
A simulation study of the shape of β precipitates in Mg–Y and Mg–Gd alloys
Acta Mater.
A thermodynamic database for zirconium alloys
J. Nucl. Mater.
A quantitative comparison between C0 and C1 elements for solving the Cahn–Hilliard equation
J. Comput. Phys.
Evaluating zirconium–zirconium hydride interfacial strains by nano-beam electron diffraction
J. Nucl. Mater.
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