Equilibrium and diffusion in coherent multilayers

A thermodynamic treatment is presented of isothermal phase equilibria and diffusion in coherent planar multilayers. For two-component systems, coherent two-phase equilibrium compositions can often be generated using the familiar procedure in which a line is constructed doubly tangential to the coherent free energy functional(s). Only in special cases, however, are the phase equilibria so generated independent of the average composition (for unsupported multilayers) or the substrate effective composition (for multilayers coherently attached to a substrate). The common tangent construction is extended to ternary solution systems, and conditions for which the coherent ternary equilibria are independent of the average or substrate compositions defined.

The formal thermodynamic aspects of diffusion in binary and ternary solids are reviewed, and the expected effects of coherency strain on the diffusional homogenization process outlined. The diffusion formalism can often be extended to account for coherency strains simply by substituting the coherent free energy density for its incoherent counterpart. In ternary and higher order systems, it is always possible to choose an initial condition that is free of strain; it is not in general possible to maintain this strain-free condition during the course of diffusional homogenization. The general effect of strain energy on ternary diffusion is to rotate the diffusion eigenvectors such that the strain is reduced. Some sample calculations are presented for the system Cu---Au---Ag.     

Phase relations in the Nb-Pd-Hf-Al system

In support of computational design of coherent aluminide strengthened Nb alloys, the phase equilibria in a series of six Nb-Pd-Hf-Al alloys were investigated using scanning electron microscopy (SEM) and x-ray diffraction (XRD). All alloys were heat treated at 1200°C for 200 h, while three alloys were also heat treated at 1500°C for 200 h. Three-phase equilibria and two-phase equilibria were observed in two alloys at both temperatures. Two more alloys also exhibited three-phase equilibria at 1200 °C, while one alloy exhibited two-phase equilibria at 1200 °C. A final alloy revealed four-phase equilibria at both temperatures, confirming the presence of a unique tie-tetrahedron in the Nb-Pd-Hf-Al system. Based on the characterization of phases by SEM and XRD, the Nb-Pd-Hf-Al partial phase diagram at 1200 °C has been constructed. Such phase diagrams provide the basis for designing precipitation-strengthened Nb-based superalloys.     

Mg-Sn-Y三元系富Mg角500℃等温截面的测定

采用合金法,利用XRD、SEM-EDS测定一系列Mg-Sn-Y三元合金在500℃下富Mg角处相平衡关系及各相平衡成分,建立Mg-Sn-Y三元系在500℃下富Mg角处的等温截面相图。结果表明:Mg-Sn-Y三元系富Mg角处存在Mg2Sn、MgSnY、Sn3Y5和Mg24+xY54种化合物与α-Mg固溶体平衡,从而构建3个三相区和4个两相区;Sn在α-Mg基体中的固溶度为2.5%～3.9%(摩尔分数),Y在α-Mg基体中的固溶度为1.1%,但二者不能同时固溶到α-Mg基体中,同时Sn3Y5相中大约可以固溶3.6%～4.1%的金属Mg;由于MgSnY和Sn3Y5等一些高熔点化合物在高温下能够稳定存在,使得Mg-Sn-Y体系有可能成为一种潜在的新型耐热镁合金。     

Particle size effects on the coherent phase equilibria of binary nanoparticles

The phase stability of isolated, radially symmetric nanoparticles of a binary system that exhibits a miscibility gap was analyzed by constructing coherent phase diagrams which account for both the surface stress (([^(T)]_s )(\hat T_s ))and the second-order compositional dependence of the lattice parameter (η _cc ). Although the elastic stress field in a two-phase coherent particle with a concentric core-shell structure is heterogeneous and nonhydrostatic at equilibrium, the appropriate free energy extremized for equilibrium could be expressed as a function solely of the temperature (ϕ), composition (c), and effective pressure (P), which are homogeneous in each phase at equilibrium. The construction of coherent phase diagrams in the three-dimensional ϕ-c-P space showed that the miscibility gap can be either extended or reduced by decreasing the particle radius, depending on the sign of[^(T)]_s h_cc\hat T_s \eta _{cc} , and that the tie-lines lie in thec-P plane.     

AuCu3-亚格子系统的合金基因Gibbs能配分函数和平衡全息网络相图（英文）

以AuCu3-亚格子系统为例,介绍3项发现:第一,迄今阻碍金属材料科学进步的第三大障碍是研究者们习惯用平衡均匀转变的思维方式认识温度极其缓慢变化的合金相变实验现象,然后以实验现象的错误认识为选择信息,建立Gibbs能函数和所谓的"平衡相图";第二,AuCu3-型亚格子系统的平衡全息网络相图可用来描述与成分和温度有关的合金基因排列结构和各种热力学性质的系统相关性;第三,每个合金的平衡转变都是均匀的单相转变,不是非均匀的双相转变,存在一条没有有序相和无序相共存区的单相相界线,相界线顶点成分和温度远偏离AuCu3化合物临界点的计量成分和温度。     

The practical use of phase diagrams

Phase diagrams are graphical representations of the states of equilibrium available to materials systems and the influence on the equilibrium states of changes in composition, temperature, and pressure. They are governed by the phase rule, which relates the number of degrees of freedom that an equilibrium has to the number of components and the number of phases occurring in the system. Phase equilibria in unary, binary, ternary, and quaternary systems are discussed, and phase and stability diagrams are applied to an understanding of vacuum sublimation, the removal of copper from liquid lead, the galvannealing of steel, tool steels, the production of Ti-B-C composites, stainless steels, the production of SiC and Si₄N₃, Al₂O₃-SiC composites, and processing of the superconducting oxide YBa₂Cu₃O_7-x.     

The practical use of phase diagrams

Phase diagrams are graphical representations of the states of equilibrium available to materials systems and the influence on the equilibrium states of changes in composition, temperature, and pressure. They are governed by the phase rule, which relates the number of degrees of freedom that an equilibrium has to the number of components and the number of phases occurring in the system. Phase equilibria in unary, binary, ternary, and quaternary systems are discussed, and phase and stability diagrams are applied to an understanding of vacuum sublimation, the removal of copper from liquid lead, the galvannealing of steel, tool steels, the production of Ti-B-C composites, stainless steels, the production of SiC and Si₄N₃, Al₂O₃-SiC composites, and processing of the superconducting oxide YBa₂Cu₃O_7-x.     

A thermogravimetric investigation of the Fe-Ni-S system from 700 to 900°C

A thermogravimetric method employing H₂-H₂S-Ar gas mixtures was used to measure the thermodynamic properties and phase equilibria of the Fe-Ni-S system. The equilibrium sulfur pressures in the ternary monosulfide phase and an adjacent two-phase region were measured between 700 to 900°C. Isothermal sections at 700, 750, 800, and 850°C are constructed as well as stability diagrams for the same temperatures.     

Coherent phase equilibria in a bending film

The theory of coherent phase diagrams is expanded to phase transformations in films. The misfit between the phases dictates their layer arrangement with an interface parallel to film surfaces. Then, the transformational self-strain leads to the film bending. The phase equilibrium analysis shows that the polymorphic transformation in a single component system proceeds with the thermodynamic hysteresis. However, the reversible change of the two-phase equilibrium with temperature is possible, if the phase fractions are maintained within a range (1/2, 2/3). The character of polymorphic phase transformations in a binary system depends on the concentration of a second component. The quasi-single component transformation takes place at small concentrations. The reversible coherent phase transformation should be observed at the concentration larger than the critical one. The irreversible transformations take place at intermediate concentrations. The irreversible transformations begin with the formation of a finite fraction of the second phase. Then, the two-phase system evaluates with temperature to the final two-phase state that transfers discontinuously to a single-phase state. The concentration diagram does not coincide with the phase diagram. The concentrations in the transforming phases change non-monotonously with temperature.     

Alloy compositions of metallic glasses and eutectics from an idealized structural model

Using an idealized atomic packing model for metallic glasses (MGs), we predict the binary alloy compositions structurally favorable for the stability of MGs. Such compositions are proposed to be the likely candidates to obtain MGs. The predicted compositions coincide very well with the experimental glass-forming range for a large number of systems. The relationship between these compositions and the eutectic compositions in the equilibrium phase diagrams is established. Direct correlations are found for symmetric eutectic systems, especially for deep eutectics, whereas off-eutectic MGs are suggested for asymmetric eutectic systems. Rationales from the structural standpoint are also presented to explain the increased frequency of appearance of eutectics at certain composition ratios (the Stockdale/Hume-Rothery puzzle).     

An Approximate Analysis of the External Oxidation of Ternary Alloys Forming Insoluble Oxides. II: Low Oxidant Pressures

The construction and the properties of three-dimensional diagrams showing the regions of stability of the various compounds, which can form as a result of the oxidation of ideal ternary A–B–C alloys by a single oxidant at a constant temperature (kinetics diagrams) are examined for oxidant pressures insufficient to oxidize all possible alloys within the system (low oxidant pressures). For the calculation it is assumed that the various oxides do not dissolve into each other and do not form double oxides and that the alloy has an ideal behavior, while internal oxidation of the most-reactive components is disregarded. The range of meaningful oxidant pressures is divided into six intervals, which correspond to the formation of different types of scales. The simplified two-dimensional (2D) kinetics diagrams presented are obtained by projecting the appropriate three-dimensional (3D) lines of equilibrium between the alloy and the various oxides on the base triangle, which gives the composition of the system in terms of the three metal components only. The kinetics diagrams are correlated with the corresponding equilibrium phase diagrams for the same quaternary A–B–C–O systems.     

Phase equilibria in the Co-rich Co-Al-W-Ti quaternary system

We determined phase equilibria in the Co-rich Co-Al-W-Ti quaternary system at a temperature range between 900 °C and 1200 °C with a close attention to the thermodynamic stability of the γ′-Co₃(Al, W, Ti) (L1₂) phase, based on micro-structure observation and electron microprobe analysis on bulk alloy samples heat-treated for periods up to 2000 h. In the quaternary system the single phase field of γ′ extends from the Co-Ti binary edge to a composition of Co-5Al-8.5W-8Ti (in at.%) at 900 °C. At the tip of the single phase field, the γ′ phase is in equilibrium with the γ-Co (A1), Co₂AlTi (L2₁) and Co₃W (D0₁₉) phases. The constructed vertical section of phase diagram between Co-9.4Al-9.6W and Co-16.5Ti indicates that there is a narrow composition range around Co-4.5Al-5.4W-7.5Ti in which the γ single phase field exists at high temperatures above 1200 °C and two-phase of γ+γ′ is thermodynamically stable at low temperatures below 1100 °C.     

Deviation from local equilibrium at migrating phase interfaces

The effects of four causes of deviation from local equilibrium at migrating phase interfaces are illustrated with molar Gibbs energy diagrams and using the sharp interface model. The conditions during solidification and melting are compared. In particular, it is examined when the composition of a growing phase can move inside the two-phase field.     

An Approximate Analysis of the External Oxidation of Ternary Alloys Forming Insoluble Oxides. I: High Oxidant Pressures

The general properties of the isothermal thermodynamic phase diagrams for ternary A–B–C alloys exposed to a single oxidant at high temperatures are examined, assuming, for simplicity, the formation of insoluble p-type oxides, an ideal behavior of the alloy, and a common value of the self-diffusion coefficient for the alloy components, independent of the alloy composition, but disregarding the possibility of internal oxidation. Simplified two-dimensional phase diagrams are produced by projecting the lines and points of equilibrium of the complete three-dimensional diagrams on the base triangle, which gives the composition of the system in terms of the metal components only. Similar kinetics diagrams concerning the nature and the growth kinetics of the scales formed on ternary alloys as functions of the bulk substrate composition are also calculated for oxidant pressures above the stability of the oxides of all the alloy components and some of their general features are examined. The kinetics diagrams for ternary alloys exposed to a single oxidant contain seven regions, three of which correspond to the areas of stability of each single oxide, three to the possible mixtures of two oxides, and one to a mixture of all three oxides.     

Mg-Zn-Al三元系富镁角在300°C的相平衡(英文)

采用平衡合金法,利用X射线衍射、扫描电镜及能谱分析,确定Mg-Zn-Al三元系富镁角300°Cα-Mg相平衡关系和相组成。结果表明:在富镁角存在3个三相区:α-Mg+Mg17Al12(γ)+Al5Mg11Zn4(φ),α-Mg+Mg32(Al,Zn)49(τ)+Al5Mg11Zn4(φ)和α-Mg+MgZn+Mg32(Al,Zn)49(τ)。与α-Mg相平衡的金属间化合物都具有很大的成分范围,并非呈线性。同时Zn和Al都能够溶解在α-Mg固溶体中,使金属间化合物达到相平衡。     

Glossary for Binary Phase Diagram Reactions

All the cases of two-phase and three-phase equilibria in isobaric binary phase diagrams are presented. New names are proposed for currently unnamed equilibria.     

A generalized thermodynamic treatment of phase equilibria in coherent multilayers

We present a generalized way of treating phase equilibria in coherent planar multilayers. A correct recognition of elastic stress and strain components as thermodynamic potentials or densities is crucial for the use of the criteria for intrinsic stability as well as for the applicability of the conventional method of common tangent construction and the Gibbs phase rule. It is shown that a method analogous to the conventional common tangent construction exists in thermodynamic density subspaces for which some of density variables are held constant. In a thermodynamic density subspace withm _s density variables fixed, a common tangent construction can be made between the extremized free energies for systems of (m_s+l)-phase coexistence in order to satisfy the thermodynamic equilibrium conditions of systems with more thanm _s+1 coexisting phases. This method is applied to a coherent binary system configured as plane-parallel plates to demonstrate that equilibrium states with more than two coexisting phases cannot be thermodynamically stable in the binary multilayer system under certain mechanical loading conditions.     

Experimental Investigation of the Ti–V Side in the Ti–V–Sn Ternary System

Phase relations in the Ti–V–Sn system are of great importance for design of aerospace titanium alloys. However, reported Ti–V–Sn ternary phase diagrams present great differences. The isothermal section of the Ti–V side in the Ti–V–Sn system at 1073, 1173, and 1273 K was established using equilibria alloys. There are 11 two-phase equilibria and 3 three-phase equilibria, 9 two-phase equilibria and 3 three-phase equilibria, and 9 two-phase equilibria and 3 three-phase equilibria in the isothermal section at 1073, 1173, and 1273 K, respectively. In addition, remarkable ternary solubility in some binary compounds was detected, e.g., up to 21.18 and 22.23 at.% V in Ti₃Sn and Ti₂Sn, respectively, at 1273 K.     

Practical applications of phase diagrams in continuous galvanizing

Zn-Fe based high-order phase diagrams have found a wide range of applications in continuous galvanizing. With the development of computer software DEAL (DetermineEffectiveALuminum), the Zn-rich corner of the Zn-Fe-Al phase diagram is being used daily for scientific interpretation of bath assays. Computer software PAL (PredictAluminumLevel), also developed by Teck Cominco Metals Ltd., assesses transient equilibria between the steel substrate and the liquid galvanizing alloy for the estimation of Fe dissolution and Al consumption in galvanizing. Aluminum deportment in galvanizing baths has been scientifically described based on the fact that bath assays corresponding to different locations and depths of a galvanizing bath formed one tie-line in the liquid-Fe₂Al₅ two-phase field of the Zn-Fe-Al phase diagram. Zn-Fe based high-order phase diagrams also afford a better understanding of the mechanisms for a number of industrial phenomena. These practical applications of the Zn-Fe based phase diagrams are detailed in the article. This article is based on a presentation made at the Galvanizers Association Ninety-Seventh Meeting held in Lexington, KY, from Oct 17 to 19, 2005.