Ginzburg-Landau-Type Multiphase Field Model for Competing fcc and bcc Nucleation

Abstract

We address crystal nucleation and fcc-bcc phase selection in alloys using a multi-phase-field model that relies on Ginzburg-Landau free energies of the liquid-fcc (face centered cubic), liquid-bcc (body centered cubic), and fcc-bcc sub-systems, and determine the properties of the nuclei as a function of composition, temperature and structure. With a realistic choice for the free energy of the fcc-bcc interface, the model predicts well the fcc-bcc phase-selection boundary in the Fe-Ni system. PACS numbers: 64.60. 64.60.My, 82.60.NFreezing of undercooled liquids often starts with the nucleation/formation of metastable (MS) crystalline phases. In agreement with Ostwald's step rule, atomistic simulations imply that the first crystal structure to form is the one, whose free energy is the closest to the free energy of the liquid [1]. In alloys this represents a multiphase multi-component solidification problem. To date, the most efficient method used for addressing such problems is the multi-phase-field theory (MPFT) [2]. It is, however, only as accurate as the free energy functional it relies on. Early versions [2] of the MPFT predicted that the third phase inevitably appears at the interface between two bulk phases, a behavior originating from the specific free energy surface assumed. A recent version of MPFT eliminated the third phase entirely at the interface [3]. This is not always in agreement with real systems: Atomistic simulations for the Lennard-Jones (LJ) system show that although the stable phase is fcc, small nuclei have a bcc structure, and even the larger fcc crystallites have a bcc-like layer at the solid-liquid interface [1,4], results also born out by the classical density functional theory (CDFT) [5]. These findings accord with the theoretical prediction of Alexander and McTague that in simple liquids the formation of bcc structure is preferred [6]. Further simulations for the LJ system imply that varying the pressure at fixed temperature, the bcc/fcc phase ratio can be tuned in small clusters [7]. Since preference for MS phase nucleation is quite general, it is desirable to work out microscopic models that can handle the structural aspects of phase selection during nucleation.In this Letter, we present such a microscopic model for competing fcc and bcc structures. The MPFT is supplemented with a free energy that is based on the GL expansion of the two-phase free energies [8][9][10], and considers thus the structural aspects of multiphase solidification. Our approach is unique in that it combines crystal structure with thermodynamic and interfacial data of real systems. In this respect our MPFT is more flexible than recent CDFT approaches [11], which, in turn, provide a more detailed description of the solid-solid interface. Herein, we apply the GL-free-energy based MPFT to predict phase-selection in the Fe-Ni system.The standard MPFT form of the grand free energy of a binary system relative to the initial liquid is:(1) The differential operator on the right hand side has the required symmetries [2...

Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: an overview

“…Atomistic simulations for the LJ system have verified that small heterophase fluctuations have the metastable bcc structure, and even larger clusters of the stable fcc structure have a bcc interface layer [214], while the ratio of the two phases can be tuned by changing the pressure [215]. Composite bcc-fcc nuclei have also been predicted by continuum models [216]. Two-stage nucleation has been reported in systems that have a metastable critical point in the undercooled liquid (including solutions of globular proteins [217]); the appearance of the crystalline phase is assisted by dense liquid droplets, whose formation precedes and helps crystal nucleation [218].…”

Multiphase phase field theory for temperature-induced phase transformations: Formulation and application to interfacial phases

“…In contrast to the other multiphase models [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15], in the developed model each of the PTs for large K ij can be described by a single order parameter without explicit constraints. It allows us to utilize analytical solutions [9] for the interface between two phases propagating in the x−direction, including its profile, width δ, and velocity c. Due to equivalence of all equations for P 0 ↔ P i and P j ↔ P i PTs, the analytical solution for a propagating with velocity c interface [9] for the P i P j interface solutions are:…”

“…Theories with a constraint. Traditional multiphase theories [9][10][11][12][13][14][15][16][17][18] include constraint η i = 1 applied to all phases. It can be explicitly excluded for two phases only, in contrast to the case with three phases for polar order parameters [8,[24][25][26].…”

“…The second multiphase approach is developed within the community working on multiphase solidification (e.g., in eutectic and peritectic systems) and grain growth [9][10][11][12][13][14][15][16][17][18]. It operates with n + 1 order parameters η i satisfying constraint η i = 1, similar to phase concentrations.…”

“…We also assume that P 0 is the high-temperature phase and, consequently, ∆s < 0. Then instability conditions (12)- (13) reduce to P 0 → P 1 : θ < θ 0 c ; θ 0 c := (a∆s θ e − B * )/(a∆s − A * ); a∆s < A * ;…”