A new analysis for the determination of ternary interdiffusion coefficients from a single diffusion couple

Concentration profiles that develop in a ternary diffusion couple during an isothermal anneal can be analyzed directly for average ternary interdiffusion coefficients. A new analysis is presented for the determination of average values for the main and cross-interdiffusion coefficients over selected regions in the diffusion zone from an integration of interdiffusion fluxes, which are calculated directly from experimental concentration profiles. The analysis is applied to selected isothermal diffusion couples investigated with α (fcc) Cu-Ni-Zn alloys at 775 °C, β (bcc) Fe-Ni-Al alloys at 1000 °C, and γ (fcc) Ni-Cr-Al alloys at 1100 °C. Average ternary interdiffusion coefficients treated as constants are calculated over composition ranges on either side of the Matano plane and examined for the diffusional interactions among the diffusing components. The ternary interdiffusion coefficients determined from the new analysis are observed to be consistent with those determined by the Boltzmann-Matano analysis at selected compositions in the diffusion zone. The ternary interdiffusion coefficients are also employed in analytical solutions based on error functions for the generation of concentration profiles for the selected diffusion couples. The generated profiles are a good representation of the experimental profiles including those exhibiting uphill diffusion or zero-flux plane (ZFP) development for the individual components. Uncertainties in the values of the interdiffusion coefficients calculated on the basis of the new analysis are found to be minimal.     

Interdiffusion in the Fe-Pt System

Diffusion-couple experiments are conducted in the Fe-Pt system. The phase boundary compositions of the phases measured in this study are found to be different than the compositions published previously. In the γ-FePt solid solution, the interdiffusion coefficient increases with the Pt content up to 25 at. pct Pt. Fe is the faster diffusing species in this phase. The trend in the interdiffusion coefficient is explained with the help of calculated driving force for diffusion. To reduce errors, the average interdiffusion coefficients are calculated in the FePt and FePt₃ compounds.     

Thermodynamic calculation for alloy systems

The formulas for calculating the activity coefficient, γ _i, in the binary system and the activity coefficient of a solute at infinite dilution, γ _i ⁰ , as well as interaction parameters, ε _i ⁱ and ε _i ^j , in a metallic melt have been proposed on the basis of Miedema et al.’s model for the estimation of the heat of formation of binary alloys. The formulas can be used in both solid and liquid solutions and have been used in the calculation of both binary and ternary alloy systems taken from 75 elements. The results were compared with the experimental values.     

Ternary diffusion: The (2~D) matrix of an ideal solid solution

The elements (~D _ik ³ ) of the (²~D) matrix of an interdiffusing ideal ternary solid solution have been determined for Manning’s random alloy model, and the relationships among the elements and the tracer diffusion coefficients (D_i*’s) of the atom species have been examined. It is shown that for physically realistic choices of tracer diffusion coefficients a quasi-linear relationship exists between the ~D _ik ³ of the ith atom species and its atom fraction over the entire composition triangle. It is also, shown that a ~D _ii ³ diagonal element will never be negative for any physically realistic values of the D_i*’s. Finally, the implications of the equality ~D ₂₂ ³ = ~D ₂₂ ³ are demonstrated.     

Stability of an alloy/oxide interface with oxygen ions being the dominant diffusing species in the oxide scale

The criterion for interfacial stability between an alloy and its oxide was formulated by Wagner [J. electrochem. Soc.103, 571 (1956)] for the specific case when the cations are the diffusing species in the oxide scale. However, for the oxidation of (Pb, In) and (Pb, Sn) alloys, oxygen in the predominant diffusing species in the oxide scales. The criterion formulated by Wagner is no longer applicable. In the present study, the criterion was formulated for the case when oxygen ions are the dominant diffusing species, following the approach used by Wagner. The criterion was used to determine the interfacial stability of (Pb, In)/In₂O₃. In order to determine the (Pb, In)/In₂O₃ interfacial stability, it is necessary to know the interdiffusion coefficient of (Pb, In) and the intrinsic diffusion coefficient of oxygen in In₂O₃. Intrinsic diffusion coefficients for In₂O₃ are not available and are estimated from the parabolic rate constants for the oxidation of (Pb, In) alloys. Calculations using the interdiffusion coefficient of (Pb, In) from the literature and the estimated intrinsic diffusivity of oxygen indicated that the (Pb, In)/In₂O₃ interface should be planar.     

The activity coefficient of oxygen in binary liquid metal alloys

The Wagner model with one energy parameter,h, for describing the effect of alloying elements on the activity coefficients of nonmetallic solutes in liquid metals is extended to have two energy parameters,h ₁andh ₂. The validity of both the Wagner one-parameter equation and the newly derived two-parameter equation is tested using data available in the literature for twelve ternary metal-oxygen systems. In order to have consistent thermodynamic data, all the relevant binary, as well as the twelve ternary metal-oxygen systems are evaluated using the same thermodynamic values for the reference materials which were used in carrying out the experimental measurements. It is found that the twoparameter equation is capable of quantitatively accounting for the compositional dependences of the activity coefficients of oxygen in all twelve ternary systems while the Wagner one-parameter equation is not. A correlation between the Wagner parameter,h, and the thermodynamic properties of the respective binary metal-oxygen and binary metals systems is found, from which the value of this parameter may be predicted without referring to any ternary data. Accordingly, the two-parameter equation is more useful in evaluating ternary experimental data while the Wagner one-parameter equation in connection with the correlation betweenh and binary data is capable of predicting ternary data without any experimental investigation in the ternary region. Based on the one-parameter and the two-parameter equations, theoretical equations for the first-order and second-order free energy interaction parameters,(∈ ₀ ^j )_sand(ρ ₀ ^j )_s, are derived in terms of the model parameters. The values of(∈ ₀ ^j )_s and(ρ ₀ ^j )_s for all the systems are derived and are found to vary linearly with the reciprocal of temperature. Furthermore, linear relationships between these two interaction parameters and their slopes with 1/T are found, from which the temperature dependence of the interaction parameters may be estimated in the absence of experimental data.     

Interaction of W<Subscript>2</Subscript>B<Subscript>5</Subscript> with Me<Superscript>IV,V</Superscript>C carbides

It is established that the W₂B₅-Me^IV,VC join in the W-B-C-Me^IV,V quaternary systems is quasibinary, while polythermal joins are described by eutectic phase diagrams. It is shown that the eutectic composition and temperature correlate with the melting points of carbides and the Me ^d C content varies from 50 mol % in the W₂B₅-VC system to 4–6 mol % in the W2B5-TaC system. The existence of Me ^d C-Me ^d B₂-W₂B₅ ternary eutectic systems promising for the development of new ceramics is proved.     

Distribution coefficients in dilute binary and ternary alloys

In most treatments of the solidification of dilute multicomponent alloys, the distribution coefficients (k ₀) are assumed to retain their binary values. Thermodynamic calculations indicate that for alloys containing up to a few percent solute,k ₀ will be approximately constant. However, large solute interaction effects have been reported for certain tin-based ternary alloys, which would make the above procedure of doubtful validity. In this paper, a brief survey pointing out the limitations and inaccuracies inherent in standard methods for the measurement of distribution coefficients is given. The determination ofk ₀ for several tin-based ternary alloys is then described. With careful technique, the large interaction effects reported earlier are not confirmed. Maximum variations ink ₀ of 20 pct within the concentration range 0 to 2 wt pct were found, compared with up to 300 pct variation reported elsewhere. It is concluded that the use of binary distribution coefficients for dilute multicomponent alloys is a reasonable approximation.     

Correlation of the hot-hardness with the tensile strength of 304 stainless steel to temperatures of 1200°C

It is shown that the ultimate tensile strength, σ, of 304 stainless steel can be correlated with hot-hardness measurements at temperatures up to 1200°C using the expression σ = (H/3.0)(n/0.2n) ⁿ whereH is the diamond pyramid hardness number andn is the strain hardening coefficient. The strain hardening coefficient was obtained from a Meyer’s hardness coefficient at the room temperature test condition and for test temperatures up to about 0.5T _m , from the empirical relationshipn =k/λ. Herek is a constant equal to approximately 0.2 microns and λ is the subgrain dimension of the deformed specimen as obtained by transmission electron microscopy. Formerly with General Electric Company, Cincinnati, Ohio     

AlScN films prepared by alloy targets and SAW device characteristics

In this paper,we reported a surface acoustic wave(SAW) device prepared and optimized by piezoelectric films containing AIN,AIScN(Sc-20 at%) and AIScN(Sc-30 at%) by reactive magnetron sputtering using Al and AISc alloy targets.We calculated the material intrinsic electromechanical coupling coefficient k_t² of AlScN(Sc-20 at%) and AlScN(Sc-30 at%) which are much better than AIN.It can be explained by the lattice softening.Furtherly,the results were confirmed by transmission el...     

Thermodynamic data of the formation of titanium nitride in iron

The solubility of titanium nitride in liquid iron is described by the solubility product log([%Ti][%N])₁ = ?17040/T + 6.40 reported by Turkdogan. The solubility in δ ferrite measured by Kunze is log<[%Ti][%N])_δ = ?17205/T + 5.56. Combining the solubilities in both phases and the solubilities of nitrogen the distribution equilibrium of titanium can be derived. It is characterized by the thermodynamic distribution coefficient k^δ/I_o,Ti = [%Ti]_δ/[%Ti] = 0.40. By zone melting and secondary ion mass spectrometry of the titanium distribution k^δ/I_Ti = 0.53 was measured. An analysis of all known data led to k^δ/I_o,Ti = 0.40…0.50. Measurements of the TiN solubility in austenite by heat treatment were not significant. They led to distribution coefficients between 0.07 and > 1. By zone melting in a carburizing atmosphere a distribution coefficient k^γ/I_Ti = 0.12 was measured. From a thermodynamic analysis performed by Ohtani et al., from the TiN solubility in the melt, and from the solubilities of nitrogen k^γ/I_o,Ti = 0.13 was deduced. Basing on the distribution equilibria of titanium and nitrogen and on the measured temperature dependence the solubility product log([%Ti][%N])_γ = ?15000/T + 4.06 was obtained.     

Interdiffusion and Growth of the Phases in CoNi/Mo and CoNi/W Systems

Deleterious topological-closed-packed (tcp) phases grow in the interdiffusion zone in turbine blades mainly because of the addition of refractory elements such as Mo and W in the Ni- and Co-based superalloys. CoNi/Mo and CoNi/W diffusion couples are prepared to understand the growth mechanism of the phases in the interdiffusion zone. Instead of determining the main and cross-interdiffusion coefficients following the conventional method, we preferred to determine the average effective interdiffusion coefficients of two elements after fixing the composition of one element more or less the same in the interdiffusion zone. These parameters can be directly related to the growth kinetics of the phases and shed light on the atomic mechanism of diffusion. In both systems, the diffusion rate of elements and the phase layer thickness increased because of the addition of Ni in the solid solution phase, probably because of an increase in driving force. On the other hand, the growth rate of the μ phase and the diffusion coefficient of the species decreased because of the addition of Ni. This indicates the change in defect concentration, which assists diffusion. Further, we revisited the previously published Co-Ni-Mo and Co-Ni-W ternary phase diagrams and compared them with the composition range of the phases developed in the interdiffusion zone. Different composition ranges of the tcp phases are found, and corrected phase diagrams are shown. The outcome of this study will help to optimize the concentration of elements in superalloys to control the growth of the tcp phases.     

Modeling of solidification microstructures in concentrated solutions and intermetallic systems

Most of the theoretical models for the predictions of solidification microstructure and solute segregation are based on the assumption mat the solute distribution coefficient,k, is independent of temperature. For concentrated alloys and for alloys near intermetallic compounds,k may vary significantly with temperature. A theoretical analysis which shows the necessary modifications in the theoretical models which must be made ifk varies with temperature is developed. It is shown that for phase diagrams with linear liquidus and solidus segments, many of the results derived with constantk can be used if the solute distribution coefficientk is replaced by a modified parameterk* which includesk as well as the derivative ofk with composition. The application of the model to concentrated alloys and to compositions near intermetallic phases is discussed. It is shown that the variation ink with temperature can significantly alter the composition dependence of dendritic microstructural scales and change the solute segregation profiles in solidified alloys.     

Modeling of solidification microstructures in concentrated solutions and intermetallic systems

Most of the theoretical models for the predictions of solidification microstructure and solute segregation are based on the assumption mat the solute distribution coefficient,k, is independent of temperature. For concentrated alloys and for alloys near intermetallic compounds,k may vary significantly with temperature. A theoretical analysis which shows the necessary modifications in the theoretical models which must be made ifk varies with temperature is developed. It is shown that for phase diagrams with linear liquidus and solidus segments, many of the results derived with constantk can be used if the solute distribution coefficientk is replaced by a modified parameterk* which includesk as well as the derivative ofk with composition. The application of the model to concentrated alloys and to compositions near intermetallic phases is discussed. It is shown that the variation ink with temperature can significantly alter the composition dependence of dendritic microstructural scales and change the solute segregation profiles in solidified alloys.     

A Derivation of a Quadratic Activity Coefficient <Emphasis Type="Italic">vs</Emphasis> Composition Relationship in a Quaternary System, <Emphasis Type="Italic">A-B-C-D</Emphasis>

In the literature, no direct derivation exists of the quadratic activity coefficient vs composition relationships for a quaternary system with high solute concentrations. Such relations for a ternary system (1-2-3) were derived by Darken by extending the results of a binary system (1-2), introducing a new concept of “hypothetical system” (2-3). To present a better scheme to find the activity coefficient-composition relations for multicomponent systems, derivations are made for a quaternary system A-B-C-D in the current work. Using a MacLaurin series expansion, the (Raoultian) activity coefficient, ln γ _i , of each component is equated with a quadratic expression of mole fractions (x), involving the activity coefficient at zero concentration ( g_i⁰ ) \left( {\gamma_{i}^{0} } \right) and nine interaction coefficients (ε). Subsequently, with the help of a Gibbs–Duhem equation, followed by a comparison of coefficients, most preceding 9 × 4, i.e., 36 interaction coefficients are eliminated, leaving behind only three self- and three ternary interaction coefficients, which are enough to express the activity coefficient vs composition relationships for the solutes B, C, and D, as well as for the solvent A. Setting the mole fraction x _D  = 0, the preceding expressions establish the same relations as proposed by Darken for the ternary system A-B-C. The derivation also clarifies how the quadratic concentration terms accompany the first-order interaction coefficients, not the second-order ones. Applications of the derived relations to determine simultaneously the activity coefficients g_i⁰ \gamma_{i}^{0} and the interaction coefficients ε in a new way in some iron- and steelmaking systems are presented. A new data on interaction coefficients in liquid iron at 1873 K (1600 °C), e_\textV^\textV = - 6. 1, \varepsilon_{\text{V}}^{\text{V}} = - 6. 1, has been generated through such an application.     

Evaluation of the activity and molecular form of bi in cu smelting slags: Part I. ternary silicate slags

The thermodynamic behavior of bismuth in the chemical systems associated with copper processing is not well understood. This study was designed to further the understanding of the physical chemistry of bismuth in slags that have similar compositions to those found in copper extractive metallurgical processing. The silicate system investigated was the FeO-Fe₂O₃-SiO₂ ternary system in which bismuth was dissolved using an isopiestic experimental technique. Bismuth vapor pressures of 1 • 10⁻⁵ atm and 7.5 • 10⁻⁴ atm were used, and the silicates were equilibrated with this vapor at temperatures of 1458 K and 1523 K. In these experiments, the slag composition was varied such thatP _{O 2} ranged from 10⁻¹² to 10⁻⁸ atm. Bismuth was found to enter the silicate slag in both neutral and oxidic molecular forms. The oxidic form identified was that of BiO. The data suggest that the activity coefficient of neutral bismuth, γ_Bi, is dependent on the solubility of that species in slag, even at the low concentrations observed in this study. It has been hypothesized, based on the large diameter of neutral Bi, that only a limited number of sites are available to accommodate neutral Bi, and that as the limit is approached γ_Bi increases significantly. That hypothesis is shown to be consistent with the experimental results obtained in the present work as well as the results obtained by other investigators.     

Kinetic model for reduction of iron oxide in molten slags by iron-carbon melt

Rate of reduction of iron oxide in iron and steelmaking slags by mass contents of dissolved carbon (>3%) in molten iron depends upon activity of FeO, temperature, mixing of bulk slag and other experimental conditions. A general kinetic model is developed by considering mass transfer of FeO in slag, chemical reaction at gas-metal interface and chemical reaction at gas-slag interface, respectively, as the three rate controlling steps. A critical analysis of the experimental data reported in literature has been done. It is shown that in the case of slags containing mass contents of less than 5% FeO, the reduction of FeO is controlled by mass transfer of FeO in slag plus chemical reaction at gas-metal interface; when slags contain more than 40% FeO, the reduction of FeO is controlled by chemical reaction at gas-metal interface plus chemical reaction at gas-slag interface; at intermediate FeO mass contents (between ～ 5 and 40% FeO), the reduction of FeO is controlled by all three steps, namely, mass transfer of FeO in slag, chemical reaction at gas-metal interface and chemical reaction at gas-slag interface. The temperature dependence of rate constant for the gas-slag reaction is obtained as: In k₂ = –32345.4(&6128)/ T + 19.0(&3.42); σ_lnk2,1/T = &0.3. where k₂ is expressed in mol m^-2 s^-1 bar^-1. The mass transfer coefficient of iron oxide in bulk slag is found to vary in the range 1.5 × 10^-5 to 5.0 × 10^-5 m/s, depending upon the slag composition as well as experimental conditions.     

Chemically based model to predict distribution coefficients in the Cu-LIX 65N and Cu-KELEX 100 systems

A chemically based thermodynamic model to predict the distribution coefficients have been developed for the Cu-LIX 65N and Cu-KELEX 100 systems. The predictive model makes use of the aqueous phase cupric sulfate complex stoichiometric stability constant expressed as its degree of formation, their extraction mechanism, and the equilibrium constant for the extraction reaction. The distribution coefficient of copper can be predicted by the equationK _d =κ₁α₀ϕ², where ϕ is the ratio of the equilibrated organic concentration to the equilibrated hydrogen ion concentration in the aqueous phase,k ₁ is the effective equilibrium constant containing the quotient of the activity coefficients of the reacting species and α₀ is the degree of formation of the free cupric ion in the equilibrated aqueous phase. Excellent agreement between the experimental data and the predicted values was obtained.