Mathematical modeling and numerical simulation of the formation and growth of a two-phase layer during diffusion in ternary system

A system of partial differential equations was derived to describe diffusion in the two-phase region of a ternary alloy by assuming that the mixture concentration in an infinitesimally small volume can be changed by external fluxes over the boundaries of this volume. A numerical procedure was developed for solving the system of partial differential equation (PDE) and treating the problem with the moving boundaries between various single-phase or two-phase layers. The mathematical model was applied for numerical simulation of the formation and growth of the single-phase and two-phase layers, developed in ternary alloy specimen, through diffusion from the surface to the core.     

Mechanical properties and lattice misfit of γ/γ′ strengthened Co-base superalloys in the Co–W–Al–Ti quaternary system

A continuous γ/γ′ two-phase field has been identified extending between the ternary Co–Al–W system to the binary Co–Ti system. The lattice misfits of two phase γ/γ′ alloys from the Co–Al–W–Ti quaternary system were measured by X-ray diffraction and found to be positive and vary linearly with composition. Differential scanning calorimetry measurements showed that the solidus and liquidus temperatures decrease from the W and Al rich end to the Ti rich end, whilst the γ′ solvus temperature increases. Long term heat treatments identified the occurrence of discontinuous precipitation at the grain boundaries in many of the alloys studied. The high temperature strength and creep resistance of the quaternary alloys in the intermediate composition range surpassed those of the binary and ternary alloys.     

The 1100 °C isothermal section of the Ti-Co-Si ternary system

The 1100 °C isothermal section of the Ti-Co-Si ternary system has been determined, for which 24 single-phase fields, 42 two-phase fields, and 22 three-phase fields were detected. Ten ternary phases were detected; these include eight previously reported in the literature plus two previously reported, H(Ti₄CoSi₄) and J(Ti₃Co₂Si). The I phase, whose crystallographic data have not been reported before, is found to be orthorhombic with a=0.7961 nm, b=0.7048 nm, and c=0.5467 nm. Except for K and H, all ternary phases show considerable composition ranges, especially the G′ and V phases.     

The 1100 °C isothermal section of the Ti-Co-Si ternary system

The 1100 °C isothermal section of the Ti-Co-Si ternary system has been determined, for which 24 single-phase fields, 42 two-phase fields, and 22 three-phase fields were detected. Ten ternary phases were detected; these include eight previously reported in the literature plus two previously reported, H(Ti₄CoSi₄) and J(Ti₃Co₂Si). The I phase, whose crystallographic data have not been reported before, is found to be orthorhombic with a=0.7961 nm, b=0.7048 nm, and c=0.5467 nm. Except for K and H, all ternary phases show considerable composition ranges, especially the G′ and V phases.     

Primary study of the isothermal phase diagram of the Cu2O-Al2O3-SiO2 ternary system at 1150℃ in air

The isothermal phase diagram of the Cu2O-Al2O3-SiO2 ternary system at 1150℃ was reported for the samples which were prepared from sol-gel method and quenched by water after being heated at 1150℃ for 12 h. Based on the conventional X-ray powder diffraction (XRD) and in situ high-temporature XRD quantitative analysis, in addition to scanning electron microscopy measurement, the phase identification was achieved. Combining the deduction from the component phase diagrams of the binary systems using the phase equilibrium theorem, the primary isothermal phase diagram was plotted over the composition area Cu2O-mullite-SiO2. In this area, the approximate composition areas of two two-phase regions and one three-phase region, (L2 + Cr), (L2 + M), and (L1 + L2 + Tr), were determined. Moreover, the precise composition areas of both of the three-phase regions (L2 + Cr + M) and (L2 + M + A) were determined according to the results of conventional and in situ high-temperature XRD quantitative analysis by Rietveld method.     

Einfluß der Zusammensetzung von Messinglegierungen auf die Entzinkung und interkristalline Korrosion in Ammoniumchloridlösung

Influence of the composition of brass on dezincification and intercrystalline corrosion in ammonium chloride solutions The intercrystalline corrosion of As-containing α-brass is not restricted to a defined brass composition but is found also with brass types containing, in addition, AL and Sn. In the case of two-phase alloys of this type the β-phase is preferentially corroded, in Particular when the brass contains As and the β-phase forms a coherent network. On the basis of the results obtained a hypothesis has been derived as to mechanism the partial processes; this hypothesis is confirmed by result obtained with macroelements of As-containing and As-free brass. According to this hypothesis the intercrystalline corrosion of α- brass can be ascribed to the inhibition by As of the dezincification, so that no regions more anodic than grain boundaries are formed. This is why the boundaries are attacked preferentially.     

Reaction diffusion of MoSi2 and Mo5SiB2

To establish quantitative basis for oxidation-protective coating of Mo–Si–B ternary alloys by MoSi₂, phase transformations in MoSi₂ vs. Mo₅SiB₂ diffusion couples have been studied. Two layers are formed on reaction diffusion at temperatures between 1400 and 1600 °C: a single-phase layer of Mo₅Si₃ and a two-phase layer consisting of Mo₅Si₃ and MoB. The growth obeys the parabolic low for both the layers, and the rate constants of the two layers are found to be approximately equal. The interdiffusion coefficient in the T₁ layer has also been evaluated. The microstructural evolution in the diffusion zone is modeled in terms of mass conservation, as well as that of a Mo–9Si–18B two-phase alloy coated with MoSi₂ reported previously [Intermetallics 12 (2004) 407].     

Three-Dimensional rendering and phase analysis of the CaO-Al2O3-SiO2 system

A program for rendering ternary phase equilibria in 3-point perspective was developed using Microsoft Visual Basic. Algorithms for rendering and phase analysis are described. With the software, liquidus, subliquidus, and solidus surfaces can be generated, and the three-dimensional object can be rotated to any angle for viewing. The operator can select a composition on the top-view 2-dimensional projection, and surfaces pertinent to an isoplethal study will be displayed. A phase analysis plot is simultaneously displayed wherein relative proportions and compositions of phases are enumerated for any chosen temperature. Phase analysis calculations were based on methods of analytical geometry using best-fit polynomials for liquidus surfaces and their contact boundaries. Phase analysis involved separation into one of five possible modes of phase evolution during cooling.     

Oxidation of multicomponent two-phase alloys

The high-temperature corrosion behavior of two-phase alloys presents a number of differences compared to that of single-phase alloys. These differences are mainly a consequence of the limitations that the presence of two phases impose on the diffusion of the alloy components. In this review, it is shown that the exclusive scale formation of the more stable, slow-growing oxide is more difficult on a two-phase alloy, requiring a higher concentration of the more reactive alloy component than for a corresponding single-phase alloy. The main types of corrosion behavior for binary two-phase alloys are also considered, showing that if diffusion in the alloy is slow the scale structure will closely reflect that of the starting material. When diffusion in the alloy is not negligible, the scale structure becomes similar to what forms on single-phase alloys. The oxidation of two-phase ternary alloys is shown to be even more complex than the two-phase binary alloys. The principal added complexity compared to the binary alloys is that diffusion in the ternary alloys may also occur in the presence of two metal phases, as a result of an extra degree of freedom in the ternary system. The oxidation behavior of two-phase ternary alloys is discussed in the context of a number of recent experimental results.     

Combined ab initio and experimental study of A2 + L21 coherent equilibria in the Fe–Al–X (X = Ti,Nb, V) systems

A combined theoretical and experimental study of ordering and phase separation in α-Fe alloys in the Fe–Al–X (X = Ti, Nb, V) systems is presented. The theoretical part is dedicated to the assessment of the BCC-based phase equilibrium diagram in the iron-rich zone of the ternary systems via a truncated cluster expansion, through the combination of Full-Potential-Linear augmented Plane Wave (FP-LAPW) electronic structure calculations and of Cluster Variation Method (CVM) thermodynamic calculations in the irregular tetrahedron approximation. The stability and the solid solubility of Al in the Fe₂X Laves phases are also included in the discussion of the ternary BCC ground state diagrams. The approach was employed in order to explore particular vertical sections of the ternary systems where a coherent two-phase microstructure can be generated with an optimal combination of volume fraction of Fe₂AlX (L2₁) Heusler type precipitates and Al content in the α-Fe (A2) matrix. The results indicate that in the Fe–Al–Ti and Fe–Al–V systems there are two kinds of phase separations of the BCC phase, (A2+ L2₁) and (B2 (FeAl structure) + L2₁). A tie-line separates both two-phase fields that shrinks and moves toward the Fe-X binary system while its direction remains almost parallel as the temperature increases. Selected experiments were performed on three alloys of the Fe–Al–Ti system belonging to vertical sections that contain this tie-line. The microstructure and composition of the matrix and precipitate phases were characterized by transmission electron microscopy (TEM) and Energy Dispersive Spectroscopy (EDS), the theoretical predictions were borne out.     

Three-Dimensional rendering and phase analysis of the CaO-Al2O3-SiO2 system

A program for rendering ternary phase equilibria in 3-point perspective was developed using Microsoft Visual Basic. Algorithms for rendering and phase analysis are described. With the software, liquidus, subliquidus, and solidus surfaces can be generated, and the three-dimensional object can be rotated to any angle for viewing. The operator can select a composition on the top-view 2-dimensional projection, and surfaces pertinent to an isoplethal study will be displayed. A phase analysis plot is simultaneously displayed wherein relative proportions and compositions of phases are enumerated for any chosen temperature. Phase analysis calculations were based on methods of analytical geometry using best-fit polynomials for liquidus surfaces and their contact boundaries. Phase analysis involved separation into one of five possible modes of phase evolution during cooling.     

Calculating models of mass action concentrations for structural units or ion couples in RbCl-H2O binary system and RbCl-RbNO3-H2O ternary system

Thermodynamic models of calculating mass action concentrations for structural units or ion couples in RbCl-H2O binary and RbCl-RbNO3-H2O ternary strong electrolyte aqueous solutions were developed based on the ion and molecule coexistence theory at 298.15 K. A transformation coefficient is needed to compare the calculated mass action concentration and the reported activity because they are obtained at different standard states and concentration units. The results show that the transformation coefficients between the calculated mass action concentrations and the reported activities of the same structural units or ion couples in RbCl-H2O binary and RbCl-RbNO3-H2O ternary strong electrolyte aqueous solutions change in a very narrow range. The transformed mass action concentrations of structural units or ion couples in RbCl-H2O binary system are in good agreement with the reported activities. The transformed mass action concentrations of RbCl and RbNO3 in RbCl-RbNO3-H2O ternary solution are also in good agreement with the reported activities, aRbCl and , with different total ionic strengths as 0.01, 0.05, 0.1, 0.5, 1.0, 1.5, 2.0, 3.0 and 3.5 mol/kg, respectively. All those results mean the developed thermodynamic model of strong electrolyte aqueous solutions can reflect structural characteristics of RbCl-H2O binary and RbCl-RbNO3-H2O ternary strong electrolyte aqueous solutions and the mass action concentration also strictly follows the mass action law.     

Oxidation of Two Ternary Fe–Cu–5 at.% Al Alloys in 1 atm of Pure O2 at 800°C

The oxidation of two two-phase ternary Fe–Cu–Al alloys containing about 5 at.% aluminium, one Fe-rich and one Cu-rich, has been studied at 800°C under 1 atm O₂. The Fe-rich alloy (Fe–15Cu–5Al) shows two parabolic stages, with a large decrease of the parabolic rate constant after about 2 hr. The presence of 5 at.% Al reduces significantly the oxidation rate of this alloy with respect to a binary Fe-Cu alloy of similar composition by forming an external alumina scale. Moreover, the addition of 15 at.% Cu is able to reduce the critical aluminium content needed to form alumina scales with respect to binary Fe–Al alloys. On the contrary, the Cu-rich Fe–85Cu–5 Al alloy presents a single parabolic stage and forms a thick and porous external scale composed of an outermost layer of copper oxides and an inner region containing a mixture of copper and Fe–Al oxides, coupled to the internal oxidation of iron and aluminium. As a result, the oxidation of the Cu-rich ternary alloy at 800°C is much faster than that of the Fe-rich ternary alloy.     

Stability of B2 ordered phase in the Ti-rich portion of Ti–Al–Cr and Ti–Al–Fe ternary systems

The stability region of the B2 phase at 1000°C in the Ti-rich part of the Ti–Al–Cr and Ti–Al–Fe ternary systems are investigated by energy dispersive spectroscopy (EDS) and transmission electron microscopy (TEM) using two-phase alloys and diffusion couples. It is established that the critical boundaries of the A2/B2 continuous ordering transition are functions of both the Al and Fe or Cr contents, and the phase equilibria between the α₂ and the β and between the β and FeTi (B2) phases are strongly affected by the A2/B2 order–disorder transition. By extrapolating these ternary data to the Ti–Al binary and using the Bragg–Williams–Gorsky approximation a metastable A2/B2 ordering boundary is postulated to exist at 1000°C in the vicinity of 23.5 at%Al in the Ti–Al binary system.     

The Zr-Ni-Cr system at 1000 °C in the ZrCr2-ZrNi-Ni-Cr region

The isothermal section of the Zr-Ni-Crternary system was determined at 1000 °C in the ZrCr₂-ZrNiNi-Cr region. The nature and composition of the phases involved in the different equilibria were determined by metallographic analysis, electron probe microanalysis (EPMA), and x-ray powder diffraction (XRD) analysis. Substitution of Cr by Ni in the Laves phase ZrCr₂ is possible in large quantities resulting in the formation of ternary Laves phases. However formation of the ZrNi₂ compound was not observed. This substitution reduces the content of Zr and induces transformations from C15 to C14 and to the C15 structure through two two-phase regions.     

The isothermal section of the Er–Fe–Sb ternary system at 773 K

The phase relation of the Er–Fe–Sb ternary system at 773 K has been investigated mainly by means of X-ray powder diffraction with the aid of optical microscopy and differential thermal analysis. This section consists of 12 single-phase regions, 22 two-phase regions and 11 three-phase regions. A ternary compound Er₆FeSb₂ has been confirmed.     

700℃ ISOTHERMAL SECTION OF THE Pd-Ru-Gd PARTIAL PHASE DIAGRAM

The 700℃ isothermal section of the Pd-Ru-Gd partial ternary phase diagram containing more palladium than ruthenium was determined by X-rag diffraction analysis and optical microscopy. The section consists of seven single-phase regions,eleven two-phase regions and five three-phase regions. A new ternary compound Pd2RuGd2,which exhibits a tetragonal structure with a=0.7492 nm and c=0.7445 nm, has been found.     

Dynamical phase analysis along pseudobinary line Ni2Cu---Ni2CuSn

The ternary system Ni---Cu---Sn includes some phases with interesting mechanical properties. The first group are the spinodal alloys in the ‘Cu corner’, the second is a group of intermetallics with large lattice parameters such as DO₃, DO₁₉ and 2H. The investigation of the pseudobinary phase diagram Ni₂Cu---Ni₂CuSn was undertaken to enable in future the preparation of samples with controlled phase composition for further study of mechanical properties of such intermetallics and of related two or, three phase composites.

The solvus line and other boundaries were revealed with the help of the so-called dynamical methods based on the measurements of physical quantities during controlled linear change of temperature. The data are related to measurements after annealing at various temperatures and also to determination of the structure after cooling to room temperature. The main methods used were X-ray diffraction (XRD) and analytical electron microscopy (AEM). The results are related elsewhere to measurement of the electrical resistivity and to the determination of solidus and liquidus temperatures by means of differential thermal analysis (DTA).     

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.     

Ir–Hf–Zr ternary refractory superalloys for ultra-high temperatures—Phase and microstructural constitution

A phase and microstructural evaluation of Ir–Hf–Zr ternary alloys with a composition below 30 mol% (Hf + Zr) was conducted by microstructural observation using scanning electron microscopy (SEM), composition analysis using electron probe microscopy analysis (EPMA), and phase identification using X-ray diffraction analysis (XRD). Partial isothermal sections of the Ir–Hf–Zr ternary system close to the Ir corner at 1800 °C and 2000 °C were determined. Research revealed that the f.c.c. and L1₂–Ir₃(Hf, Zr) two-phase regions, which are shown in the Ir–Hf and Ir–Zr binary systems, were connected from the Ir–Hf side to Ir–Zr side in the Ir–Hf–Zr ternary system at Hf + Zr contents of less than 25 mol%. The L1₂–Ir₃Hf and L1₂–Ir₃Zr phases were fully soluble with each other. An Ir₃(Hf, Zr)/Ir(Hf, Zr) two-phase structure was found in the Ir–15Hf–15Zr alloy with the Hf + Zr contents of 30 mol%. The potential of the Ir–Hf–Zr ternary alloys as ultra-high-temperature structural materials is discussed from the viewpoints of the microstructure and the lattice misfit between the f.c.c. and the L1₂ phases.