Multiphase binary diffusion in infinite and semi-infinite media: Part II. On the numerical calculation of the rate constants for formation of product phases

The equations describing multiphase binary diffusion derived in the preceding article are arranged in an order such that the unknowns in the equations are expressed as two series of sequential functions of the corresponding two independent variables. The conditions to be satisfied between one series of functions and the other are given, and a numerical method for solving this type of equations of two unknowns is developed. For the numerical calculation of the parabolic rate constants for formation of product phases, data of average interdiffusion coefficients, partial molal volumes, and phase boundary compositions are required for an infinite medium. For a semi-infinite medium, in addition to these data, information on the equilibrium surface composition and the ratio of mole transfers of the two components at the surface is required.     

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.     

A practical solution for the diffusion equations in binary and multicomponent systems with constant intrinsic diffusion coefficients

A practical solution for the diffusion equations in binary and ternary systems is presented which leads to a prediction of the concentration-penetration curves, diffusion paths, and the values for the intrinsic and interdiffusion fluxes. The model is very simple and gives insight into a variety of diffusion phenomena and zero-flux planes. The model has been applied to systems in which the intrinsic diffusion coefficients are constant and which, as far as ternary systems are concerned, are thermodynamically ideal. Although not mathematically exact, the results agree within the expected experimental accuracy with exact solutions presented in the literature. Even if the intrinsic diffusion coefficients are not constant, or if the ternary system is not thermodynamically ideal, the results agree semi-quantitatively with experimental results found in the literature. With some adaptations the agreement will be still better.     

Multiphase precipitation of carbides in Fe-C systems: Part I. Model based upon simple kinetic reactions

In this work, we present a theoretical scheme that allows us to follow the kinetics of the precipitation of carbides in the Fe-C system. Our scheme is based on reaction equations, which take into account all possible ways of atomic exchange between the different competing phases. Through the use of nonlinear regression methods, we have fitted the results from our model to those obtained from thermoelectric power measurements for the Fe-C system in the temperature range between 20 °C and 450 °C. By deconvolution of optimally fitted theoretical kinetics, we have deduced ranges for coexistence of the different carbides that formed and their activation energies. An adequate agreement between our results and those reported in the literature is obtained.     

Infiltration of fiber preforms by a binary alloy: Part I. Theory

During infiltration of a fiber preform by a binary hypoeutectic alloy, solid metal can form in the composite because of cooling at the fibers or at the mold wall. Contrary to the case of an unalloyed matrix, temperature, composition, and fraction solid may vary in the composite. This results in macrosegregation and microstructural heterogeneity within the composite casting. It is shown that solid metal that forms because of cooling at the fibers grows gradually behind the infiltration front, while the local temperature increases. Metal superheat, when present, serves to progressively remelt solid metal in the composite during infiltration and increases compositional and microstructural heterogeneity within the composite. General expressions are derived to describe heat, mass, and fluid flow during the infiltration process. In the case of unidirectional adiabatic infiltration driven by a constant applied pressure, a similarity method can be used to reduce the mathematical complexity of the problem. Numerical solution of the resulting equations then allows us to predict temperature, fraction solid, and composition profiles within the composite. With the further assumption of negligible thermal conduction, the problem lends itself to an analytical solution. The analysis is performed for the case of unidirectional adiabatic infiltration under constant applied pressure of 24 vol pct δ-alumina preforms by Al-4.5 wt pct Cu. Results indicate that there is significant latitude for control of macro-segregation and microstructure within cast fiber-reinforced alloys.     

On the calculations of the diffusion coefficients of oxygen and nitrogen in niobium

The Snoek peaks for oxygen and nitrogen in niobium have been remeasured using a torsion pendulum equipped with LVDT transducers connected to an oscillograph for automatic data recording. The peak temperatures were determined using a new numerical analysis of the peak shape. These new low temperature data have been combined with high temperature direct data and intermediate temperature high frequency internal friction data reported in the literature. When it is assumed that the oxygen and nitrogen] occupied octahedral positions in the niobium matrix, the diffusion coefficients for both oxygen and nitrogen atoms follow a simple Arrhenius behavior. In the temperature range of this analysis (140 to 1000°C for oxygen and 270 to 1800°C for nitrogen) the equationsD = (5.3E - 7) exp - (1.095E5)/RT andD = (2.6E - 6) exp - (1.523E5)/RT apply for the diffusion of oxygen and nitrogen, respectively. Entropy of activation calculations were made using the above activation energies and frequency factors, and the results agree well with reported values for other systems as well as with the theory of Keyes. This good agreement suggests that the assumption that oxygen and nitrogen prefer octahedral sites is probably correct.     

Multiphase Diffusion in Fe−Ni−Al System at 1000°C: I. Diffusion structures and diffusion paths

Multiphase diffusion was investigated in the Fe−Ni−Al system at 1000°C with several series of solid-solid diffusion couples assembled with selected β(bcc) and γ(fcc) alloys for the development of diffusion structures. The series of couples were characterized by a common β or γ terminal alloy joined to selected alloys of the other phase. The diffusion structures developed for the various couples exhibited interfaces of either planar, wavy, acicular, or flowery morphologies. Diffusion paths for the couples were determined by electron microprobe analysis. On the basis of isoconcentration contours in the vicinity of interfaces, the path-crossings of the (β+γ) two-phase region were found to be parallel to tie-lines for planar β/γ interfaces but were spread out into a band inclined to tie-lines for nonplanar interfaces. The band spread of path crossings for nonplanar β/γ interfaces showed compositional variations mainly on the γ-side of interfaces. G. H. CHENG formerly a Graduate Student at Purdue University     

Interface morphology development during stress corrosion cracking: Part I. Via surface diffusion

The initiation of a crack in a specimen under tensile or compressive stresses is treated from the point of view of perturbation analysis. A surface distortion is Fourier analyzed into a series of waves and the amplitude response of a single component of varying frequency is theoretically investigated. The response of the individual components yields a Griffith-type criterion for wave amplitude growth. The model is applied to alloy systems undergoing stress corrosion cracking via surface diffusion.     

Single-particle tracking: the distribution of diffusion coefficients

In single-particle tracking experiments, the diffusion coefficient D may be measured from the trajectory of an individual particle in the cell membrane. The statistical distribution of single-trajectory diffusion coefficients is examined by Monte Carlo calculations. The width of this distribution may be useful as a measure of the heterogeneity of the membrane and as a test of models of hindered diffusion in the membrane. For some models, the distribution of the short-range diffusion coefficient is much narrower than the observed distribution for proteins diffusing in cell membranes. To aid in the analysis of single-particle tracking measurements, the distribution of D is examined for various definitions of D and for various trajectory lengths.     

Morphological healing evolution of intragranular penny-shaped microcracks by surface diffusion: Part I. Simulation

An axisymmetric finite-element method is developed and applied to study spheroidization and breakup of intragranular penny-shaped microcracks by surface diffusion. As the initial aspect ratio of a microcrack increases, it undergoes different sequences of morphological change, evolving into the shapes of a spherical void, a doughnutlike channel pore, a ring with a center void, or multiple rings with or without a center void, while the evolution time continuously increases. During the process of separating into multiple rings, the subsequent separation is a relatively fast process in comparison with the first splitting event. The rate of the morphological evolution depends sensitively on the initial aspect ratio, in particular, the initial thickness, of the microcrack. The simulations also show that the spheroidization rate of a penny-shaped microcrack is much greater than the cylinderization rate of a two-dimensional (2-D) elliptic microcrack, which requires a much larger initially critical aspect ratio for splitting.     

Diffusion in the titanium-nickel system: II. calculations of chemical and intrinsic diffusion coefficients

Diffusion coefficients in the Ti-Ni system have been calculated by the aid of equations given by Sauer and Freise, and Wagner. Values for the TiNi (50 at. pct Ni) phase were found to be:D ^u (cm²/s) = 0.0020 exp - 142,000/R for the temperature range between 650 and 940°C. The heat of activation, expressed in J/mol, has an accuracy of ±6000. For the β-Ti(Ni) phase containing 6 at. pct Ni the temperature dependence of the diffusion coefficient is expressed by:D ^u (cm²/s) = 0.0688 exp - 141,000/RT. The uncertainty in the energy of activation is ±12000 J/mol. No clear variation of the diffusion coefficient with concentration could be detected. It was found that Ni is by far the fastest moving component in β-Ti(Ni), Ti₂Ni and TiNi (at least in the composition range between 50 and 53 at. pct Ni). Values ofD _Ni/D _Ti have been calculated with an equation derived by van Loo. The significance of the calculated values is critically examined. By means of a practical example it is shown that the calculated ratio of the intrinsic diffusion coefficients can be extremely sensitive to slight variations in the position of the marker interface.Diffusion coefficients in the Ti-Ni system have been calculated by the aid of equations given by Sauer and Freise, and Wagner. Values for the TiNi (50 at. pct Ni) phase were found to be:D ^u (cm²/s) = 0.0020 exp - 142,000/R for the temperature range between 650 and 940°C. The heat of activation, expressed in J/mol, has an accuracy of ±6000. For the β-Ti(Ni) phase containing 6 at. pct Ni the temperature dependence of the diffusion coefficient is expressed by:D ^u (cm²/s) = 0.0688 exp - 141,000/RT. The uncertainty in the energy of activation is ±12000 J/mol. No clear variation of the diffusion coefficient with concentration could be detected. It was found that Ni is by far the fastest moving component in β-Ti(Ni), Ti₂Ni and TiNi (at least in the composition range between 50 and 53 at. pct Ni). Values ofD _Ni/D _Ti have been calculated with an equation derived by van Loo. The significance of the calculated values is critically examined. By means of a practical example it is shown that the calculated ratio of the intrinsic diffusion coefficients can be extremely sensitive to slight variations in the position of the marker interface. This paper is based on a Thesis submitted by G. F. BASTIN in fulfillment of requirements for the degree of Doctor in Technological Sciences.     

Kinetics of the extraction of metals from deep-sea manganese nodules: Part I. the pore diffusion controlling case

The rate of dissolution of metal ions from two different deep-sea manganese nodule samples in dilute sulfuric acid solutions was studied. The addition of high concentrations of sodium and calcium salts was observed to increase the apparent kinetics of extraction and this has been interpreted in terms of the physical and chemical nature of nodules and their component minerals. The dissolution of copper and soluble cobalt was found to be very sensitive to the acid concentration and to be controlled by pore diffusion, while nickel dissolution was found to be controlled by chemical reaction. A pore model has been developed for the leaching of copper and cobalt. Formerly with the University of California, Berkeley.     

The role of diffusion in determining the controlling creep mechanisms in Al-Zn solid-solutions: Part I

This is the first of a two part paper in which the role of diffusion in determining the rate controlling creep mechanisms in binary solid solutions is investigated. Part I deals with the appropriate diffusion coefficients for climb and glide controlled creep, while Part II involves a comparison of the climb and glide creep equations with experiment. The first part begins with a review of a number of diffusion coefficients that have been proposed for creep in binary solid solutions. We show that the diffusion coefficient for climb is while that for glide is . Two other diffusion coefficients that have been used in the literature are shown to be incorrect. To experimentally verify the theoretical diffusion coefficients, the creep behavior of Al-Zn alpha solid solutions was studied over a composition range of 0 to 60 at. pct Zn. Values of and the thermodynamic factor (1 +d lnγ _i/d lnX i) are taken from the literature and used to calculate D_cland D_glas a function of composition at 360 and 410°. It is found that the thermodynamic factor strongly affects the diffusion coefficientD _gl for glide-controlled creep. When these curves forD _cl andD _gl are compared with experimental creep data for the Al-Zn system,D _gl is found to explain the decrease in creep rate observed at the miscibility gap composition. Formerly Graduate Research Assistant     

Rate of decarburization of iron-carbon melts: Part I. experimental determination of the effect of sulfur

The kinetics of decarburization of iron-carbon melts with CO-CO₂ gas mixtures were investigated at 1700 ° using the levitation technique. The influences of different experimental variables on the decarburization kinetics were determined. It was found that sulfur has a clear and reproducible retarding effect on the decarburization of iron-carbon melts; and this effect is most pronounced at sulfur concentrations in the range of 0 to 0.05 wt pct. The initial carbon concentration has no discernible effect on the decarburization kinetics. Melts containing 2.48 wt pct C and 0.92 wt pct C initially were found to decarburize at virtually identical rates until a substantial portion of the carbon was removed. The decarburization rate of a melt with a specified initial carbon content was found to remain essentially constant until the carbon content fell to a characteristic level below which the rate tended to level off. The partial pressure of CO₂ of the gas mixture has a marked effect on the decarburization kinetics. The flow-rate of the gas mixture has a small but finite effect on the rate of decarburization.