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The problem of simultaneous growth and competition of intermediate phases during reactive diffusion is formulated and solved. In this paper, we compare existing models of steady state reaction diffusion and introduce the new one basing on the bi-velocity method. We extend old problem and propose method based on material (lattice) fixed frame of reference. It allows computing the material velocity in the reacting system in which reactions at several moving interfaces occur. All reactions lead to the lattice shift due to the difference of intrinsic diffusivities and different molar volumes. The following peculiarities are taken into account: (1) the deviation from local equilibrium at all interfaces; (2) the mobilities of the components in the bulk and interphase zone; and (3) the molar volumes of the components. We show the kinetic of the reactions, the non parabolic regime, the multiphase scale growth and present the practical application of the method. [ABSTRACT FROM AUTHOR]

In this paper the Diffusion Path (DP) method is presented. The method allows for the determination of the diffusion path during a reactive diffusion process. The initial conditions of the method are the phase diagram, diffusion coefficients in the phases and initial composition of the diffusion couple. The method allows for the determination of the diffusion path—the intermetallic phases generated during the diffusion process and the order of their generation. The results of the DP method are compared with the experimental results obtained for the Ti-NiCu diffusion couples with various Ni to Cu content ratio. [ABSTRACT FROM AUTHOR]

In this paper the vacancies formation and agglomeration under centrifugal force — sedimentation will be analyzed. The evolutionary algorithm for diffusion and vacancy evolution will be shown. The model predicts the location of vacancies agglomeration during the sedimentation process — the most probably place of voids formation. The computed results will be compared with experiments in Cu–brass diffusion couple. The influence of the centrifugal force on the vacancies migration will be presented. [ABSTRACT FROM AUTHOR]

In this paper the interdiffusion process and Frenkel effect (Kirkendall porosity) are studied in Ni-Cu and NiAl-Cu systems at 1273 K. The generalized Boltzmann-Matano method is used for calculation of the intrinsic diffusion coefficients as a polynomial function in the whole composition range. The calculated interdiffusion coefficient in Cu-Ni diffusion couple decrease with increasing Ni content. The total decrease of interdiffusion coefficient exceeds two orders of magnitude. The equation for the void growth rate is obtained. The experimental results of diffusion and voids formation are analyzed. The voids radii is estimated and compared with experiments. [ABSTRACT FROM AUTHOR]

The complete understanding of the phenomenological process related to the position of the Matano and Kirkendall plane in multicomponent system is still lacking. In this paper some aspects of Matano and lattice plane migration are studied by means of numerical simulations. New approach to calculate intrinsic diffusion coefficients in binary Ni–Au system and ternary Cu–Fe–Ni and Co–Fe–Ni is presented. Diffusivities were calculated using modified Boltzmann–Matano method. Self diffusion coefficients of the elements in diffusion couple must be known as well as position of the Kirkendall plane after the experiment to use this method. Diffusion profile was calculated using presented equations. Calculations compared with experimental data prove validity of the presented approach. This work proves that intrinsic diffusion coefficients should be given by the concentration function, not the exact value. [ABSTRACT FROM AUTHOR]

The Poisson equation is used to calculate the drift velocity in the two-dimensional diffusion couple. This approach is based on the bi-velocity (Darken) method which combines the Darken and Brenner concepts that the volume velocity is essential in defining the local material velocity in multicomponent mixture at non-equilibrium. As an example the arbitrary binary system is considered. It is shown that (1) the two dimensional calculations should be applied with the stochastization method and (2) the drift term in mass conservation law does not affect the calculations. [ABSTRACT FROM AUTHOR]

Abstract: In this work the competition between Kirkendall effect caused by intrinsic diffusion coefficients and vacancy generation and evolution is discussed. The proposed approach is based on the bi-velocity (Darken) approach which combines the Darken and Brenner concepts that the volume velocity is essential in defining the local material velocity at non-equilibrium. The model is formulated for arbitrary systems however for discussion only the binary system is presented. The distributions of (1) components, (2) vacancy and (3) voids radius over the distance are presented. [Copyright &y& Elsevier]

Abstract: The phenomenological Boltzmann–Matano (B–M) analysis allows approximating the concentration dependent interdiffusion coefficients related to infinite binary diffusion couple experiments. However, the complete understanding of the phenomenological process related to the Kirkendall plane position coupled with the B–M method is still lacking. A purpose of the present study is to generalize the Boltzmann–Matano analysis and propose a method of estimating the unique intrinsic diffusion coefficients of components from the experimental concentration profile and the positions of Matano and Kirkendall planes in multicomponent systems. The proposed physico-chemical approach is used to approximate the intrinsic diffusivities in a single phase of Ni–Pd binary and Cu–Fe–Ni ternary diffusion couples. [Copyright &y& Elsevier]

A bi?velocity method is used to simulate the element?concentration evolution in the diffusion zone during the aluminization of a Rene?80 nickel?based superalloy. The basis of the method and the detailed calculations are presented. The model enables the prediction of the formation of intermetallic phases and multi?phase layers. The entropy?production rate can also be calculated, and it is shown that it might serve as a basis to indicate the interphases. The proposed model is applied to model the diffusion in a ternary Al?Cr?Ni system. The results of the calculations are compared with experimental element?concentration profiles, measured for the Rene?80 alloy subjected to aluminization by CVD. [ABSTRACT FROM AUTHOR]

Abstract: The understanding of the phenomenological process related to the position of the Kirkendall plane is still lacking. In this study some aspects of Kirkendall and lattice plane migration in binary diffusion couples are studied by means of numerical simulations by the bi-velocity method. The simulations are performed using constant and composition dependent intrinsic diffusion coefficients. The results show that the position and stability of the Kirkendall plane as well as a number of other interesting features can be predicted using the entropy–density curve calculated by the bi-velocity method. [Copyright &y& Elsevier]

Abstract: The Bi-velocity Phase Field Method (BPFM), which allows a quantitative description of diffusion mass transport in multi-phase materials, is presented and applied to simulate formation of a diffusion zone in aluminized MAR-247. The method links the bi-velocity Darken approach with the phase-field model in which the diffusion zone is quantitatively characterized by phase volume fractions. Accordingly, composition dependent diffusivities in a two-phase region are defined. The present method allows calculation of element-distribution profiles and volume fractions of the phases across a diffusion couple. Hence, a diffusion path can be determined for three component systems and, besides, the average stress field and entropy production during diffusion can be calculated. An example of the application of BPFM in the modeling of high-temperature reactive diffusion in an AlCrNi ternary system is presented. The simulated results are compared with experimental data obtained in aluminization of MAR-247 by the chemical vapor deposition process. [Copyright &y& Elsevier]

Abstract: An approach to the problem of interdiffusion in multicomponent–multiphase system is presented. It couples bi-velocity (Darken) method, in which the mass conservation and volume continuity laws are considered, with the phase field model and is a generalization of the model based on the diffusion equation implemented in Dictra. The proposed bi-velocity phase field method allows computing the kinetics of the process, diffusive mass transport in every phase, concentration profiles, diffusion paths, and Kirkendall effect. The example problem presented here is a fundamental problem concerning the reactive diffusion at high temperatures in Al–Cr–Ni ternary system. It is a starting point that will allow modeling the coating composition to reduce interdiffusion and stabilize a desired microstructure, i.e., to extend coating life time. [Copyright &y& Elsevier]

In this work we solve the problem of the simultaneous growth and competition of intermediate phases. The crystal lattice as the frame of reference for diffusion is defined, based on the volume continuity equation. It permits the computation of the material velocity in a system in which reactions at several moving interfaces occur. All reactions lead to the lattice shift due to the difference of diffusivities and molar volumes. We simulate the CVD process followed by annealing in a non-oxidizing atmosphere. The computed kinetics and concentration profiles are compared with experimental results. [ABSTRACT FROM AUTHOR]

In the paper the mathematical model describing the gravity induced sedimentation process in multicomponent system is presented. The model is based on the generalized interdiffusion method (bi-velocity method): (1) the volume continuity equation, (2) equation of motion, (3) Cauchy stress tensor and (4) Nernst Planck flux formulae. The method is applied to simulate the sedimentation processes of the selenium isotopes and the ternary InSbGe system. [ABSTRACT FROM AUTHOR]

Abstract: The bi-velocity method allows quantitative description of the complex diffusion mass transport process for an unlimited number of elements. The method deals with (1) composition dependent diffusivities, (2) different partial molar volumes of components, (3) the stress field during the diffusion process and (4) entropy production. It allows for calculation of the time evolution of the components’ concentration profiles for non-ideal systems when thermodynamic activity data is available. Applications of the model describing interdiffusion in a ternary Cu–Fe–Ni system under the stress field is presented. [Copyright &y& Elsevier]

A theory of stress and elastic deformation during interdiffusion is presented. It is shown that it is consistent with the general Darken analysis and linear irreversible thermodynamics. Special consideration is given to local entropy production. An expression is derived for the rate of entropy production during interdiffusion and practical computations presented of the internal energy and entropy densities and the entropy production. It is shown that the entropy produced by the diffusion of mass is positive, [image omitted], and it does not depend on the frame of reference. The results span the gap between the Darken method, linear irreversible thermodynamics and treatments by Larche, Cahn and Stephenson. [ABSTRACT FROM AUTHOR]

Abstract: The bi-velocity concept of separation of diffusion and drift flows allows one to combine the convection and dissipative contributions to the mass and energy fluxes. It is shown that the bi-velocity method combined with statements of the first law of thermodynamics and the Euler’s and Liouville theorems is consistent with linear irreversible thermodynamics. We apply the derived formalism in modeling the two-dimensional sedimentation between Bi and Sb and demonstrate the satisfactory agreement with experimental data. The method allows modeling the sedimentation of isotopes in solids. [Copyright &y& Elsevier]

In the present work surface roughness of copper samples after mechanical surface preparation processes (polishing, grinding, sand-blasting) and after exposure in non-oxidizing (high purity argon) and oxidizing (air) conditions at 700 and 750 °C have been evaluated by contact profilometer and by fractal analysis. Quantitative description of surfaces that are hidden from the sight of conventional methods is possible by fractal analysis, e.g. in the present work, the roughness of Cu 2 O/Cu boundary has been evaluated under oxide layer. It has been pointed out, that in both oxidizing and non-oxidizing conditions initial surface roughness of copper is affected by annealing at high temperature: at 700 °C surface roughness of polished and ground samples increased but at 750 °C decreased in comparison to surface roughness prior to heat treatment. For sand-blasted samples in both oxidizing and non-oxidizing conditions the lowest values of surface roughness parameters were obtained at 750 °C. It is then concluded that two phenomena are responsible for such effect: chemical reaction between copper and oxygen and surface diffusion of copper due to high temperature of annealing. Unlabelled Image • Surface of copper samples had been polished, ground and sand-blasted. • Surface roughness had been evaluated by contact profilometer and fractal analysis. • There are differences in roughness after annealing in oxidizing and non-oxidizing conditions at different temperatures. • Changes in roughness are caused by chemical reaction between Cu and O and surface diffusion of Cu at high temperature. [ABSTRACT FROM AUTHOR]

The experimental concentration profile in Ti-NiAl diffusion couple after diffusion at 1173 K for 100 h. Determination of diffusion coefficients in a ternary multiphase system is discussed as the data on the approximation method allowing to determine the diffusion coefficients in such systems are currently missing in the literature. In the present work, the mass conservation approach (MCA) for calculation the diffusivities in individual phases with the linear differences in composition is proposed. The diffusion data can be determined exclusively based on the experimental results. The method is implemented and validated for the ternary Ti-NiAl system, in which three intermetallic phases are stable. [ABSTRACT FROM AUTHOR]

In this work, the influence of three different surface preparation processes, namely polishing, grinding and sand-blasting on oxidation process of high purity Ni48Fe52 alloy (in at. %) at high temperature has been investigated. Surface roughness of differently treated samples was measured by contact profilometer. Isothermal oxidation was performed in thermogravimetric furnace at 900°, 950° and 1000 °C for 24 h in air. Cross-sections were prepared and then observed and analyzed by SEM (Scanning Electron Microscope) /EDS (Energy Dispersive Spectroscopy). It was determined that sand-blasting increases the number of dislocations and grain boundaries in the near-surface region and therefore slows down oxidation process by faster formation of mixed nickel/iron oxide that constitutes a barrier for outward iron diffusion. Internally oxidized zone is also more uniform and thicker for polished and ground samples in comparison to sand-blasted ones as a result of recrystallization of the latter and longer duration of faster oxygen solubility in the polished and ground samples. Unlabelled Image • Surface of Ni48Fe52 alloy samples were polished, ground and sand-blasted. • Surface roughness was measured by contact profilometer. • Internal FeO precipitates are fine for polished and ground samples. • Internal FeO precipitates along grain boundaries are coarse for sand-blasted samples. • Sand-blasting prior to oxidation slows down the oxidation process. [ABSTRACT FROM AUTHOR]

The interdiffusion in Ti-Ni70Cu30 diffusion couple was studied at 850 °C at times varied between 10 and 300 min. The diffusion paths for the samples were determined and the order of intermetallic phases occurring in the diffusion zone was established. It was concluded that after 10 and 30 min of diffusion annealing, four intermetallic phases formed, namely: Ti 2 Ni, TiNi, τ 1 and TiNi 3. Between 30 and 60 min, also the Ti 2 Cu phase appeared next to Ti 2 Ni phase forming the multi-zone region, which thickness almost doubled between 60 and 300 min of diffusion annealing at 850 °C. The formation of Ti 2 Cu and the multi-zone region was explained by the enrichment of Cu at the Ti 2 Ni/Ti boundary and connected with the increase in its diffusion flux. • Ti-Ni70Cu30 couple was annealed at 850 °C at times between 10 and 300 min. • after 10 and 30 min of annealing of Ti-Ni70Cu30 at 850 °C four phases formed. • between 30 and 60 min, the Ti 2 Cu appeared next to Ti 2 Ni forming two-phase region. • enrichment in Cu at the Ti 2 Ni/Ti boundary caused the formation of two-phase region. [ABSTRACT FROM AUTHOR]