Transformation In Binary Alloys Research Articles

Interface kinetics in phase-field models: Isothermal transformations in binary alloys and step dynamics in molecular-beam epitaxy

We present a unified description of interface kinetic effects in phase-field models for isothermal transformations in binary alloys and steps dynamics in molecular-beam-epitaxy. The phase-field equations of motion incorporate a kinetic cross-coupling between the phase field and the concentration field. This cross-coupling generalizes the phenomenology of kinetic effects and was omitted until recently in classical phase-field models. We derive general expressions (independent of the details of the phase-field model) for the kinetic coefficients within the corresponding macroscopic approach using a physically motivated reduction procedure. The latter is equivalent to the so-called thin-interface limit but is technically simpler. It involves the calculation of the effective dissipation that can be ascribed to the interface in the phase-field model. We discuss in detail the possibility of a nonpositive definite matrix of kinetic coefficients, i.e., a negative effective interface dissipation, although being in the range of stability of the underlying phase-field model. Numerically we study the step-bunching instability in molecular-beam-epitaxy due to the Ehrlich-Schwoebel effect, present in our model due to the cross-coupling. Using the reduction procedure we compare the results of the phase-field simulations with the analytical predictions of the macroscopic approach.

Neutron diffraction analysis and solidification modeling of Impulse-Atomized Al-36 wt%Ni

Peritectic solidification reactions appear in many metallic systems such as Al-Ni alloys. However, microstructural evolution and effects of processing parameters on phase selection during peritectic reaction are not well understood. In this paper Impulse-Atomization experiments and simulations were used to study rapid solidification of Al-36 wt% Ni particles. Secondary dendrite arm spacing measurements were used to estimate the cooling rate achieved during the solidification of these particles. The weight fractions of the phases formed in different sizes of the particles were measured using neutron diffraction technique. Solidification paths were then simulated with a model previously validated for concurrent dendritic, peritectic and eutectic phase transformations in binary alloys. Model predictions are compared to the experimental results to understand the sequence of transformations that leads to the final metallurgical state of the particles.

Modeling structural transformations in binary alloys with phase field crystals

We develop a phase field crystal model for structural transformations in two-component alloys. In particular, the interactions between components are described by direct correlation functions that are an extension of those introduced by Greenwood et al. [Phys. Rev. Lett. 105, 045702 (2010)] for pure materials. These correlation functions result in broad density modulations that can be treated with high numerical efficiency, hence enabling simulations of phase transformations between a wide range of crystal structures. A simplified binary alloy model is shown to describe the equilibrium properties of eutectic and peritectic binary alloys in two and three dimensions. The robustness and versatility of this method is demonstrated by applying the model to the growth of structurally similar and dissimilar eutectic lamella and to segregation to defects.

Gas atomization of Al–Ni powders: Solidification modeling and neutron diffraction analysis

Powerful catalysts may be produced from gas atomized Al–Ni powders. The efficiency of these catalysts depends on the amount of the different phases formed during solidification. The present paper reports on the analysis of these transformations by combining experiments and simulations. Firstly, the volume fractions of phases in gas atomized Al–Ni powders are determined for several compositions and size ranges using neutron diffraction analysis. Additionally, solidification paths are simulated with a model previously validated for concurrent dendritic, peritectic and eutectic phase transformations in binary alloys. The measurements demonstrate a strong dependence of phase fractions on particle size, as well as an inversion of trends with composition. The predictions of the model show good agreement with the measurements. Details of the simulations clarify the combined roles of the kinetics of cooling, the solute diffusion in phases as well as the growth kinetics of microstructures. These advanced interpretations should open the way to the production of further optimized catalysts.

Effect of the Compositional Strain on the Diffusive Interface Thickness and on the Phase Transformation in a Phase-Field Model for Binary Alloys

A Cahn-Hilliard phase-field—elasticity model was used to study the effect of compositional strain on the diffusive interface thickness and on the solid state phase transformations in binary alloys. Compositional strain was introduced using the Vegard’s law. Mixed order finite element analyses and analytical solutions of an infinite diffusion couple with a flat interface were used to track the phase-field interface morphology. Both analytical and numerical calculations showed a substantial rate-increasing effect of compositional strain on the interface thickness, especially for low energy barrier values. Compositional strain was found to cause substantial patterning of single precipitates during their evolution in a parent matrix and significantly change the equilibrium size of the precipitates. Results show a considerable influence of compositional strain on the coarsening kinetics of coherent precipitates.

Modeling of soft impingement effect during solid-state partitioning phase transformations in binary alloys

The soft impingement effect at the later stage of partitioning phase transformations has been modeled both for the diffusion-controlled growth model and for the mixed-mode model. Instead of the linear and exponential approximations for the concentration gradient in front of the interface used in the past, a general polynomial method of dealing with the diffusion field is proposed. The linear and exponential diffusion field approximations are two specific cases of the polynomial diffusion field approximation. The effect of soft impingement on the overall partitioning phase transformation is only related to the degree of the super-saturation in case of the diffusion-controlled growth model, while it is determined by both the growth mode and the degree of super-saturation in case of the mixed-mode model.

Finite element method for conserved phase fields: Stress-mediated diffusional phase transformation

Phase-field models with conserved phase-field variables result in a 4th order evolution partial differential equation (PDE). When coupled with the usual 2nd order thermo-mechanics equations, such problems require special treatment. In the past, the finite element method (FEM) has been successfully applied to non-conserved phase fields, governed by a 2nd order PDE. For higher order equations, the convergence of the standard Galerkin FEM requires that the interpolation functions belong to a higher continuity class. We consider the Cahn–Hilliard phase-field model for diffusion-controlled solid state phase transformation in binary alloys, coupled with elasticity of the solid phases. A Galerkin finite element formulation is developed, with mixed-order interpolation: C 0 interpolation functions for displacements, and C 1 interpolation functions for the phase-field variable. To demonstrate convergence of the mixed interpolation scheme, we first study a one-dimensional problem – nucleation and growth of the intermediate phase in a thin-film diffusion couple with elasticity effects. Then, we study the effects of completeness of C 1 interpolation on parabolic problems in two space dimensions by considering the growth of the intermediate phase in a binary system. Quadratic convergence, expected for conforming elements, is achieved for both one- and two-dimensional systems.

A new semi-analytical method for phase transformations in binary alloys

In the present paper, a new semi-analytical method is developed to cover a wide range of phase transformation problems and their practical applications. The solution procedure consists of two parts: first, determination of the position of the moving boundary named the homogenous part and second, determination of the concentration named the non-homogenous part. The homogenous part leads to a system of homogenous linear equations, based on the mathematical fact that a homogenous system has a non-trivial solution if the determinant of the coefficient matrix equals zero. This determinant leads to an ordinary differential equation for the moving boundary, and its solution leads to a closed form formula for the position of the moving boundary. The non-homogenous part transforms the governing equations to a non-homogenous linear system of equations, having three unknowns that appear in the concentration profile assumed in the beginning of the proposed method. Solution of the non-homogenous system leads to a value of these unknowns. Once these unknowns are computed, the concentration at any time and at any point can be found easily.

Phase Transformations in Binary Alloys under High Pressure

High-pressure studies of binary alloy systems display some specific features connected with composition referring as a variable parameter. In a two-component system under pressure there are observed transformations, whichdo not occur in a one-component system. In particular, there is a formation of a single phase from a mixture of two phases or a decomposition of a uniform phase into two phases of different compositions. Alloys of In and Sn with neighbouring elements give us examples of these kinds of transformations. Studies on binary alloys with application of pressure provide experimental evidences that the alloy composition or electron concentration is one of the main factor for structural phase stability following from Hume-Rothery rule.

Surface phase transitions of solids. Influence of a surface on order-disorder transformations in binary alloys

A report is given on a study of the order parameter and the order parameter fluctuations at the Fe3Al (110) surface when it approaches the order-disorder phase transformation. X-ray scattering under grazing incidence and exit was employed and gave the necessary surface sensitivity. The temperature dependence of the critical surface scattering at and around the (111) superstructure reflection was investigated. The critical behaviour is strongly influenced by the surface. The new surface exponents have been determined and are β1 = 0.75±0.06, γ‖ = –0.33±0.06, η‖ = 1.52±0.04 and ν = 0.60±0.15. The surface dominated phase transition belongs to the ordinary transition class. Scaling laws can be verified.

The Pair Approximation of the Cluster Variation Method as Applied to bcc Alloys

The phase diagram of the order-disorder transformation in binary alloys A x B 1−x with bcc structure is calculated using the pair approximation of the Cluster Variational Method (CVM). Nearest (NN) and next-nearest neighbor (NNN) interactions are considered. Two types of pairs are taken into account: short pairs for the NN and long pairs for the NNN. The results of our calculation are compared with the CVM tetrahedron approximation and Monte Carlo Simulations (MCS)

The discrete coagulation-fragmentation equations: Existence, uniqueness, and density conservation

The discrete coagulation-fragmentation equation describes the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones. These models have many applications in pure and applied science ranging from cluster formation in galaxies to the kinetics of phase transformations in binary alloys. Our results relate to existence, uniqueness, density conservation and continuous dependence and they generalise the corresponding results in [ref. 2] for the Becker-Doring equations for which the processes are restricted to clusters gaining or shedding one particle. Examples are given which illustrate the role of the assumptions on the kinetic coefficients and show the rich set of analytic phenomena supported by the general discrete coagulation-fragmentation equations.

Study of a model for the main phase transition of lipid bilayers using the Kikuchi cluster method

The statistical mechanics of Doniach's two-state lattice model for the main gel – liquid crystal phase transition of phospholipid bilayers is treated in a similar manner to order–disorder transformations in binary alloys and magnetic systems, using the cluster variation method developed by Kikuchi. Indeed, the analogy holds better for the latter system, since the entropy difference between the two states gives rise to an effective temperature-dependent field. This effective field vanishes at the first-order phase transition, whose latent heat is associated with the discontinuity in the order parameter.We use Kikuchi's approximation with the inclusion of triangle bond correlations, and pair and site probabilities in the expression for free energy. We assume that the lipid chains only interact through nearest neighbour pair potentials and that triangle correlations are important for approximate counting of allowed states. Two long-range order parameters and a short-range order parameter are introduced in the formulation of the theory. Both long-range order parameters are discontinuous at the transition temperature. Numerical results for the physical quantities are presented and discussed with respect to earlier work.

Electronic Theory of Order-Disorder Transformations in Binary Alloys

ABSTRACTA microscopic theory of order-disorder phase transitions in binary alloys is oresented. The internal energy of the system is calculated within the tight-binding Hamiltonian and long- and short-range order effects are included by solving the equations of motion for the Green's functions in the Be the lattice approximation. Results are presented for the electronic local density of states and for various values of the long- and short-range order parameters.

On the description of isothermal phase transformations in binary alloys

Two effects are considered to influence the time dependence of the transformed volume in a binary alloy: a) the impingement effect (IE), takes into account the mutual impingement of the growing particles and b) the impoverishment effect (PE), which considers the decrease of driving forces and the increase of the diffusion paths in the approximation of an impoverishing matrix. Combining both effects growth equations are obtained analogous toJohnson-Mehl-Avrami only for one-dimensional growth.

A simple model to describe isothermic phase transformations in binary alloys

A simple model to describe isothermic phase transformations in binary alloys

Theory of order-disorder and order-order transformation in binary alloys with B.C.C. lattice—III.: Atomic ordering in Fe-Al system

On the basis of the general theory developed with the account of a four-particle correlation of atoms in the tetrahedron cluster sites the atomic ordering is considered in the Fe-Al system with 0–50 at % Al. The calculated equilibrium diagram of the phases Fe 3Al(DO 3), FeAl(B2) and short-range order phases α(A2) are in agreement with the diagram proposed by Rudman. The temperature dependences of short- and long-range order parameters in the case of Fe 3Al and Fe 13Al 3 compositions are discussed.

Comments on “Experimental observations of continuous transformations in binary alloys”

Comments on “Experimental observations of continuous transformations in binary alloys”

Theory of order-disorder and order-order transformation in binary alloys with B.C.C. lattice — III. Atomic ordering in Fe-Al system

Theory of order-disorder and order-order transformation in binary alloys with B.C.C. lattice — III. Atomic ordering in Fe-Al system

Theory of order-disorder and order-order transformations in binary alloys with b.c.c. lattice—I: Free energy and phase equilibrium equations

The atomic ordering in the four phases: in the phase with short-range order and in the phases with the CuZn ( B2), NaTI ( B32) and Fe 3Al (DO 3) superstructure types is considered. For an unambiguous representation of the phase free energy on the base of a tetrahedron cluster a generalization of the Kikuchi method made by Hijmans and De Boer was used. The equations for one- and two-phase equilibrium were given.