Diffuse-interface model for rapid phase transformations in nonequilibrium systems

Pinning of crack fronts by hard and soft inclusions: A phase field study

Through tridimensonal numerical simulations of cracks propagating in material with an elastic moduli heterogeneity, it is shown that the presence of a simple inclusion can dramatically affect the propagation of the crack. Both the presence of soft and hard inclusions can lead to the arrest of a crack front. Here the mechanism leading to the arrest of the crack are described and shown to depend on the nature of the inclusion. This is also the case in regimes where the presence of the inclusion leads to a slowdown of the crack.

Kinetics of rapid growth and melting of Al50Ni50alloying crystals: phase field theoryversusatomistic simulations revisited. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2022;34:494002. [PMID: 36228604 DOI: 10.1088/1361-648x/ac9a1c] [Citation(s) in RCA: 0] [Impact Index Per Article: 0]

A revised study of the growth and melting of crystals in congruently melting Al₅₀Ni₅₀alloy is carried out by molecular dynamics (MDs) and phase field (PF) methods. An embedded atom method (EAM) potential of Purja Pun and Mishin (2009Phil. Mag.89 3245) is used to estimate the material's properties (density, enthalpy, and self-diffusion) of the B2 crystalline and liquid phases of the alloy. Using the same EAM potential, the melting temperature, density, and diffusion coefficient become well comparable with experimental data in contrast with previous works where other potentials were used. In the new revision of MD data, the kinetics of melting and solidification are quantitatively evaluated by the 'crystal-liquid interface velocity-undercooling' relationship exhibiting the well-known bell-shaped kinetic curve. The traveling wave solution of the kinetic PF model as well as the hodograph equation of the solid-liquid interface quantitatively describe the 'velocity-undercooling' relationship obtained in the MD simulation in the whole range of investigated temperatures for melting and growth of Al₅₀Ni₅₀crystals.

The hodograph equation for slow and fast anisotropic interface propagation

Using the model of fast phase transitions and previously reported equation of the Gibbs-Thomson-type, we develop an equation for the anisotropic interface motion of the Herring-Gibbs-Thomson-type. The derived equation takes the form of a hodograph equation and in its particular case describes motion by mean interface curvature, the relationship 'velocity-Gibbs free energy', Klein-Gordon and Born-Infeld equations related to the anisotropic propagation of various interfaces. Comparison of the present model predictions with the molecular-dynamics simulation data on nickel crystal growth (obtained by Jeffrey J. Hoyt et al. and published in Acta Mater. 47 (1999) 3181) confirms the validity of the derived hodograph equation as applicable to the slow and fast modes of interface propagation. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

Kinetics of solid-liquid interface motion in molecular dynamics and phase-field models: crystallization of chromium and silicon

The results of molecular dynamics (MD) simulations of the crystallization process in one-component materials and solid solution alloys reveal a complex temperature dependence of the velocity of the crystal-liquid interface featuring an increase up to a maximum at 10-30% undercooling below the equilibrium melting temperature followed by a gradual decrease of the velocity at deeper levels of undercooling. At the qualitative level, such non-monotonous behaviour of the crystallization front velocity is consistent with the diffusion-controlled crystallization process described by the Wilson-Frenkel model, where the almost linear increase of the interface velocity in the vicinity of melting temperature is defined by the growth of the thermodynamic driving force for the phase transformation, while the decrease in atomic mobility with further increase of the undercooling drives the velocity through the maximum and into a gradual decrease at lower temperatures. At the quantitative level, however, the diffusional model fails to describe the results of MD simulations in the whole range of temperatures with a single set of parameters for some of the model materials. The limited ability of the existing theoretical models to adequately describe the MD results is illustrated in the present work for two materials, chromium and silicon. It is also demonstrated that the MD results can be well described by the solution following from the hodograph equation, previously found from the kinetic phase-field model (kinetic PFM) in the sharp interface limit. The ability of the hodograph equation to describe the predictions of MD simulation in the whole range of temperatures is related to the introduction of slow (phase field) and fast (gradient flow) variables into the original kinetic PFM from which the hodograph equation is obtained. The slow phase-field variable is responsible for the description of data at small undercoolings and the fast gradient flow variable accounts for local non-equilibrium effects at high undercoolings. The introduction of these two types of variables makes the solution of the hodograph equation sufficiently flexible for a reliable description of all nonlinearities of the kinetic curves predicted in MD simulations of Cr and Si. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

Traveling waves of the solidification and melting of cubic crystal lattices

Using the phase field crystal model (PFC model), an analysis of slow and fast dynamics of solid-liquid interfaces in solidification and melting processes is presented. Dynamical regimes for cubic lattices invading metastable liquids (solidification) and liquids propagating into metastable crystals (melting) are described in terms of the evolving amplitudes of the density field. Dynamical equations are obtained for body-centered cubic (bcc) and face-centered cubic (fcc) crystal lattices in one- and two-mode approximations. A universal form of the amplitude equations is obtained for the three-dimensional dynamics for different crystal lattices and crystallographic directions. Dynamics of the amplitude's propagation for different lattices and PFC mode's approximations is qualitatively compared. The traveling-wave velocity is quantitatively compared with data of molecular dynamics simulation previously obtained by Mendelev et al. [Modell. Simul. Mater. Sci. Eng. 18, 074002 (2010)MSMEEU0965-039310.1088/0965-0393/18/7/074002] for solidification and melting of the aluminum fcc lattice.

Fast traveling waves in the phase-field theory: effective mobility approach versus kinetic energy approach

A phase-field model for small and large driving forces on solidification and melting of a pure substance or alloys is formulated. Derivations of the phase-field model are based on the effective mobility approach and on the kinetic energy approach to analyze fast phase transformation from metastable liquid to solid phase. A hodograph equation (an acceleration-velocity dependent equation of the Gibbs-Thomson type) which predicts the non-linear behavior in the velocity of the crystal-liquid interface is found at the large driving force on transformation and analyzed for different thermodynamic potentials. Traveling wave solutions of this equation are found for double-well and double-obstacle potentials. The velocity-dependent traveling waves as a function of driving force on transformation exhibit non-linearity of the solutions. Namely, in the relationship 'velocity-driving force' exists a maximum at a fixed undercooling which is very well known in the solidification of glass-forming metals and alloys. The predicted solidification velocity is quantitatively compared with the molecular dynamics simulation data obtained by Tang and Harrowell (2013 Nat. Mater. 12 507-11) for the solidification of congruently melting Cu-Zr binary alloy. The comparison confirms a crucial role of local non-equilibrium such as relaxation of gradient flow in the quantitative description of fast phase transformations.

Thermodynamics of rapid solidification and crystal growth kinetics in glass-forming alloys

Thermodynamic driving forces and growth rates in rapid solidification are analysed. Taking into account the relaxation time of the solute diffusion flux in the model equations, the present theory uses, in a first case, the deviation from local chemical equilibrium, and ergodicity breaking. The second case of ergodicity breaking may exist in crystal growth kinetics of rapidly solidifying glass-forming metals and alloys. In this case, a theoretical analysis of dendritic solidification is given for congruently melting alloys in which chemical segregation does not occur. Within this theory, a deviation from thermodynamic equilibrium is introduced for high undercoolings via gradient flow relaxation of the phase field. A comparison of the present derivations with previously verified theoretical predictions and experimental data is given. This article is part of the theme issue 'Heterogeneous materials: metastable and non- ergodic internal structures'.

Kinetic transition in the order-disorder transformation at a solid/liquid interface

Phase-field analysis for the kinetic transition in an ordered crystal structure growing from an undercooled liquid is carried out. The results are interpreted on the basis of analytical and numerical solutions of equations describing the dynamics of the phase field, the long-range order parameter as well as the atomic diffusion within the crystal/liquid interface and in the bulk crystal. As an example, the growth of a binary A₅₀B₅₀ crystal is described, and critical undercoolings at characteristic changes of growth velocity and the long-range order parameter are defined. For rapidly growing crystals, analogies and qualitative differences are found in comparison with known non-equilibrium effects, particularly solute trapping and disorder trapping. The results and model predictions are compared qualitatively with results of the theory of kinetic phase transitions (Chernov 1968 Sov. Phys. JETP26, 1182-1190) and with experimental data obtained for rapid dendritic solidification of congruently melting alloy with order-disorder transition (Hartmann et al. 2009 Europhys. Lett.87, 40007 (doi:10.1209/0295-5075/87/40007)).This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

Travelling-wave amplitudes as solutions of the phase-field crystal equation

The dynamics of the diffuse interface between liquid and solid states is analysed. The diffuse interface is considered as an envelope of atomic density amplitudes as predicted by the phase-field crystal model (Elder et al. 2004 Phys. Rev. E70, 051605 (doi:10.1103/PhysRevE.70.051605); Elder et al. 2007 Phys. Rev. B75, 064107 (doi:10.1103/PhysRevB.75.064107)). The propagation of crystalline amplitudes into metastable liquid is described by the hyperbolic equation of an extended Allen-Cahn type (Galenko & Jou 2005 Phys. Rev. E71, 046125 (doi:10.1103/PhysRevE.71.046125)) for which the complete set of analytical travelling-wave solutions is obtained by the [Formula: see text] method (Malfliet & Hereman 1996 Phys. Scr.15, 563-568 (doi:10.1088/0031-8949/54/6/003); Wazwaz 2004 Appl. Math. Comput.154, 713-723 (doi:10.1016/S0096-3003(03)00745-8)). The general [Formula: see text] solution of travelling waves is based on the function of hyperbolic tangent. Together with its set of particular solutions, the general [Formula: see text] solution is analysed within an example of specific task about the crystal front invading metastable liquid (Galenko et al. 2015 Phys. D308, 1-10 (doi:10.1016/j.physd.2015.06.002)). The influence of the driving force on the phase-field profile, amplitude velocity and correlation length is investigated for various relaxation times of the gradient flow.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

Coarse-graining for fast dynamics of order parameters in the phase-field model

In standard descriptions, the master equation can be obtained by coarse-graining with the application of the hypothesis of full local thermalization that is equivalent to the local thermodynamic equilibrium. By contrast, fast transformations proceed in the absence of local equilibrium and the master equation must be obtained with the absence of thermalization. In the present work, a non-Markovian master equation leading, in specific cases of relaxation to local thermodynamic equilibrium, to hyperbolic evolution equations for a binary alloy, is derived for a system with two order parameters. One of them is a conserved order parameter related to the atomistic composition, and the other one is a non-conserved order parameter, which is related to phase field. A microscopic basis for phenomenological phase-field models of fast phase transitions, when the transition is so fast that there is not sufficient time to achieve local thermalization between two successive elementary processes in the system, is provided. In a particular case, when the relaxation to local thermalization proceeds by the exponential law, the obtained coarse-grained equations are related to the hyperbolic phase-field model. The solution of the model equations is obtained to demonstrate non-equilibrium phenomenon of solute trapping which appears in rapid growth of dendritic crystals.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

Analysis of Ginzburg-Landau-type models of surfactant-assisted liquid phase separation

In this paper diffuse interface models of surfactant-assisted liquid-liquid phase separation are addressed. We start from the generalized version of the Ginzburg-Landau free-energy-functional-based model of van der Sman and van der Graaf. First, we analyze the model in the constant surfactant approximation and show the presence of a critical point at which the interfacial tension vanishes. Then we determine the adsorption isotherms and investigate the validity range of previous results. As a key point of the work, we propose a new model of the van der Sman/van der Graaf type designed for avoiding both unwanted unphysical effects and numerical difficulties present in previous models. In order to make the model suitable for describing real systems, we determine the interfacial tension analytically more precisely and analyze it over the entire accessible surfactant load range. Emerging formulas are then validated by calculating the interfacial tension from the numerical solution of the Euler-Lagrange equations. Time-dependent simulations are also performed to illustrate the slowdown of the phase separation near the critical point and to prove that the dynamics of the phase separation is driven by the interfacial tension.

Coarse graining for the phase-field model of fast phase transitions

Fast phase transitions under lack of local thermalization between successive elementary steps of the physical process are treated analytically. Non-Markovian master equations are derived for fast processes, which do not have enough time to reach energy or momentum thermalization during rapid phase change or freezing of a highly nonequilibrium system. These master equations provide a further physical basis for evolution and transport equations of the phase-field model used previously in the analyses of fast phase transitions.

Unconditionally stable method and numerical solution of the hyperbolic phase-field crystal equation

The phase-field crystal model (PFC model) resolves systems on atomic length scales and diffusive time scales and lies in between standard phase-field modeling and atomistic methods. More recently a hyperbolic or modified PFC model was introduced to describe fast (propagative) and slow (diffusive) dynamics. We present a finite-element method for solving the hyperbolic PFC equation, introducing an unconditionally stable time integration algorithm. A spatial discretization is used with the traditional C^{0}-continuous Lagrange elements with quadratic shape functions. The space-time discretization of the PFC equation is second-order accurate in time and is shown analytically to be unconditionally stable. Numerical simulations are used to show a monotonic decrease of the free energy during the transition from the homogeneous state to stripes. Benchmarks on modeling patterns in two-dimensional space are carried out. The benchmarks show the applicability of the proposed algorithm for determining equilibrium states. Quantitatively, the proposed algorithm is verified for the problem of lattice parameter and velocity selection when a crystal invades a homogeneous unstable liquid.

Solute trapping in rapid solidification of a binary dilute system: a phase-field study

The phase-field model of Echebarria, Folch, Karma, and Plapp [Phys. Rev. E 70, 061604 (2004)] is extended to the case of rapid solidification in which local nonequilibrium phenomena occur in the bulk phases and within the diffuse solid-liquid interface. Such an extension leads to the fully hyperbolic system of equations given by the atomic diffusion equation and the phase-field equation of motion. This model is applied to the problem of solute trapping, which is accompanied by the entrapment of solute atoms beyond chemical equilibrium by a rapidly moving interface. The model predicts the beginning of complete solute trapping and diffusionless solidification at a finite solidification velocity equal to the diffusion speed in bulk liquid.

Unconditionally gradient-stable computational schemes in problems of fast phase transitions

Equations of fast phase transitions, in which the phase boundaries move with velocities comparable with the atomic diffusion speed or with the speed of local structural relaxation, are analyzed. These equations have a singular perturbation due to the second derivative of the order parameter with respect to time, which appears due to phenomenologically introduced local nonequilibrium. To develop unconditionally stable computational schemes, the Eyre theorem [D. J. Eyre, unpublished] proved for the classical equations, based on hypotheses of local equilibrium, is used. An extension of the Eyre theorem for the case of equations for fast phase transitions is given. It is shown that the expansion of the free energy on contractive and expansive parts, suggested by Eyre for the classical equations of Cahn-Hilliard and Allen-Cahn, is also true for the equations of fast phase transitions. Grid approximations of these equations lead to gradient-stable algorithms with an arbitrary time step for numerical modeling, ensuring monotonic nonincrease of the free energy. Special examples demonstrating the extended Eyre theorem for fast phase transitions are considered.

Phase-field-crystal and Swift-Hohenberg equations with fast dynamics

A phenomenological description of transition from an unstable to a (meta)stable phase state, including microscopic and mesoscopic scales, is presented. It is based on the introduction of specific memory functions which take into account contributions to the driving force of transformation from the past. A region of applicability for phase-field crystals and Swift-Hohenberg-type models is extended by inclusion of inertia effects into the equations of motion through a memory function of an exponential form. The inertia allows us to predict fast degrees of freedom in the form of damping perturbations with finite relaxation time in the instability of homogeneous and periodic model solutions.

Solute trapping and diffusionless solidification in a binary system

Numerous experimental data on the rapid solidification of binary systems exhibit the formation of metastable solid phases with initial (nominal) chemical composition. This fact is explained by complete solute trapping leading to diffusionless (chemically partitionless) solidification at a finite growth velocity of crystals. Special attention is paid to developing a model of rapid solidification which describes a transition from chemically partitioned to diffusionless growth of crystals. Analytical treatments lead to the condition for complete solute trapping which directly follows from the analysis of the solute diffusion around the solid-liquid interface and atomic attachment and detachment at the interface. The resulting equations for the flux balance at the interface take into account two kinetic parameters: diffusion speed VDI on the interface and diffusion speed VD in bulk phases. The model describes experimental data on nonequilibrium solute partitioning in solidification of Si-As alloys for the whole range of solidification velocity investigated.