Robert Filipek

Self- and Interdiffusion in Ternary Cu-Fe-Ni Alloys

Diffusion of Cu, Fe, and Ni radiotracers has been measured in Cu–Fe–Ni alloys of different compositions at 1271 K. The measured penetration profiles reveal grain boundary-induced part along with the volume diffusion one. Correction on grain boundary diffusion was taken into account when determining the volume diffusivities of the components. When the Cu content in the alloys increases, the diffusivities increase by order of magnitude. This behaviour correlates well with decreasing of the melting temperature of corresponding alloys, as the Cu content increases. Modelling of interdiffusion in the Cu–Fe–Ni system based on Danielewski-Holly model of interdiffusion is presented. In this model (extended Darken method for multi-component systems) a postulate that the total mass flow is a sum of the diffusion and the drift flows was applied for the description of interdiffusion in the closed system. Nernst-Planck’s flux formula assuming a chemical potential gradient as a driving force for the mass transport was used for computing the diffusion flux in non-ideal multi-component systems. In computations of the diffusion profiles the measured tracer diffusion coefficients of Cu, Fe and Ni as well as the literature data on thermodynamic activities for the Cu–Fe–Ni system were used. The calculated interdiffusion concentration profiles (diffusion paths) reveal satisfactory agreement with the experimental results. Introduction Advanced materials are very often multi-component and have complex structure (gradient materials, coatings, etc.) and from thermodynamic point of view are non-ideal. Interdiffusion and interfacial reactions play important role during processing of many functional materials and limit their long term exploitation. Understanding of these processes has fundamental practical importance. There are different approaches to reach this goal. These are Onsager phenomenology [1] and Darken method [2], they differ in form of the constitutive flux formula. Up to now they were not unified and the fundamental question is whether the computed diffusivities or interdiffusion coefficients represent real material constants [3]. Situation is even more difficult because the proof of the uniqueness of the inverse Darken problem does not exist [4]. We selected relatively simple Cu-Fe-Ni system to study the interdiffusion in non-ideal alloys showing limited solubility. This system has advantage because its thermodynamic and kinetic data are fairly well known. This paper has two goals: first to measure the tracer diffusivities of all components as a function of the ternary alloy composition, second the theoretical examination of the extended Darken method (Danielewski-Holly model) for a ternary system. Experimental measurements Sample preparation and radiotracer measurements Copper, iron and nickel (99.99 pct purity) were used as starting materials. The ternary alloys (Table 1) were melted in induction furnace in argon atmosphere. Such obtained ingots were homogenized at 1273 K for 250 hour in argon ( 2 6 10 O p atm − < ). Subsequently, the alloys of different composition were reannealed at the temperature 50 C below their melting temperature for 4 hours for recrystalization. The grain size of polycrystalline alloys was measured to vary from 30 (Cu-richest alloy) to about 300 μm. Alloys in the form of the cylindrical ingots of 8 mm in diameter were cut into slices of 3 mm thick by spark erosion. One face of the specimen was polished to optical quality by standard metallographical procedure. The samples were encapsulated into Cu–Fe–Ni containers (made of nearly the same alloy which was under study), wrapped in Ta foil, and sealed in silica tubes under purified Ar atmosphere. The use of the containers guarantees that the chemical composition of the specimen was not changed during thermal treatment. The samples were subsequently annealed at 1373 K for 24 h in order to remove the mechanical stresses, which could be built up during cutting and polishing procedures and which could affect the diffusion behaviour. After the pre-annealing the blanc surface was slightly chemically polished with Syton colloidal silica slurry to remove the effects of thermal etching. Table 1. Alloy compositions in wt. % Alloy Fe Ni Cu #1 12 68 20 #2 25 50 25 #3 10 45 45 #4 28 37 35 #5 45 40 15 #6 6 24 70 #7 75 20 5 #8 10 80 10 Penetration profile measurements The radiotracer Cu (half-life 12.7 hours) was produced by neutron irradiation of a cupper chip at the research reactor in Geesthacht, Germany. The nuclear reaction was used. Its initial specific activity was about 500 MBq/mg. Due to short life-time of the isotope it was delivered to the lab in Munster in few hours after irradiation. The activated chip was dissolved in nitric acid and then diluted with double-distilled water. 63 64 29 29 Cu(n, ) Cu γ The radiotracer Fe (half-life 45 days) was also produced by neutron irradiation of iron powder at the research reactor in Geesthacht according to the nuclear reaction . The activated isotope material was dissolved in 30% HCl and then further diluted with double-distilled water. 58 59 26 26 Fe(n, ) Fe γ The radiotracer Ni (half-life 100 years) was purchased in form of a HCl solution and dissolved with double-distilled water. Each radiotracer (Cu, Fe, or Ni) was dropped as a dilute acid solution on the polished face of the samples. The samples were again encapsulated into containers, which were then wrapped into Ta foil, to avoid any undesirable contamination. The specimens were sealed in silica tubes under purified Ar atmosphere and subjected to the diffusion anneals at T = 1271 K. The temperatures were measured and controlled with Ni–NiCr thermocouples with an accuracy of about ±1 K. After the diffusion anneal the samples were reduced in diameter (at least 1 to 2 mm) by grinding on a lathe to remove the effect of lateral and surface diffusion. The penetration profiles were determined by the precision grinding sectioning technique using special abrasive mylar foils. The radioactivity of each section was determined with high efficiency using a Packard TRI CARB 2500 TR liquid scintillation counter. The penetration profiles present the plots of the normalized activity of each sections (the activity divided by the weight of the section) against the penetration depth x. The latter was calculated from the weight loss of the sample with known geometry after each sectioning step. The uncertainties of individual points on the penetration profiles, stemming from the counting procedure and the errors of the depth determination, were estimated to be typically less than 10 %. 0 5 10 15 20 25 Ni Fe Cu42.8Fe10.8Ni46.4

Neutral-carrier ion-selective electrodes assessed by the Nernst-Planck-Poisson model.

Ion-selective electrodes (ISEs) containing neutral ionophores are used in clinical, industrial, and environmental analysis. The wide range of applications requires deep theoretical description. This work concentrates on the development of the general approach to the description of electro-diffusion processes, namely, Nernst-Planck-Poisson (NPP) model to allow the description of the time-dependent responses in the case of complexation reactions occurring in the ion-selective membranes. The impact of the chemical reaction on the calibration curves and apparent selectivity of ISE is discussed. Results obtained using NPP model with time-dependent reaction are compared with those obtained with the Phase Boundary Model (PBM), as well as with the previous solutions of NPP model, using the infinite reaction rates and constant ligand concentration assumption. The validity of these assumptions is investigated and the limitations of PBM in the description of neutral-carrier ISE are discussed.

Interdiffusion Studies in Co-Fe-Ni Alloys

Interdiffusion in Co-Fe-Ni alloys was studied in 1373-1588 K temperature range. The Danielewski-Holly model was used for the description of the interdiffusion process in ternary Co- Fe-Ni diffusion couples both for the finite and infinite geometry. Using the inverse method the average intrinsic diffusivities of components in the Co-Fe-Ni system were calculated and compared with the results of the other authors. The activation energies of cobalt, iron and nickel intrinsic diffusion have been found in 1373-1588 K temperature range.

Interdiffusion under the Chemical Potential Gradient; Comparison of Onsager and Darken Models

The generalized Darken method of interdiffusion in multicomponent systems (GDM) enables obtaining an exact expression for the evolution of component distributions for arbitrary initial distributions and time dependent boundary conditions. In this work we studied the consequences of the more general formulae for the diffusional flux (Plancks equation) which permits to take into account thermodynamical driving forces. This paper is based on our studies of diffusion couples in the Cu-Fe-Ni system at 1273 K. The Cu-Fe-Ni system was chosen because it is a single phase in a wide range of compositions and because its thermodynamic properties are fairly well known. However, the solid solutions in this system are not ideal and consequently the diffusivities depend on composition. The driving force for the diffusion in such ternary system is the gradient of the chemical potential which can be calculated from the concentration profiles and using the known thermodynamical data of the system. Consequently the diffusional flux can be expressed as a function of the concentration gradients of all elements in the system, of the thermodynamical terms and of the mobilities. The diffusion paths are discussed in the light of the ternary interdiffusion coefficients and the self diffusivities with the use of generalized Darken method. A comparison of the Onsager and Darken models is presented. The results show the prospect for the future application of the GDM in the modelling of stress affected diffusion in ternary and higher solid solutions. The generalized Darken’s method A key problem in multicomponent diffusion is the prediction of the diffusion path between the two terminal alloys. For the predictive calculations, the data for the intrinsic diffusivities and/or self diffusion coefficients and their concentration dependence have to be known. Moreover the thermodynamic data for the system have to be known for the calculation of the chemical potentials for each component. The generalised Darken method (GDM) allows a complete quantitative description of the complex diffusional transport process and for the unlimited number of elements. It allows calculation of the diffusion paths when the interdiffusion coefficients are concentration dependent. The details of this model for the closed system [1] and the more general description of interdiffusion that incorporates the equation of motion can be found elsewhere [2-3]. In this paper the formulation of the initial-boundaryvalue problem for the interdiffusion in a closed system is presented. It differs from that presented recently [4,5] with the expression for the mass flux. Physical laws: 1) the law of the mass conservation of an i-th element:

Modelling of Interdiffusion and Reactions at the Boundary; Initial-Value Problem of Interdiffusion in the Open System

The application of the Danielewski-Holly model of interdiffusion for modelling of selective and concurrent oxidation of multi-component alloys is presented. This model enables prediction of the evolution of components distributions taking into account interdiffusion and the reactions at the boundary, e.g, due to the oxidation/sulphidation processes. The model is subsequently reformulated to the form suitable for numerical calculation. For illustrating its capabilities modelling of the selective oxidation of Ni-Pt alloys is presented. The results are compared with those obtained from Wagner model. Both models give exactly the same results for the longer reaction times. In Wagner model the equilibrium concentration of the elements at the boundary is reached instantly while in this model it changes with time. Consequently the model allows modelling of initial stages of oxidation.

Modelling of Oxidation of Fe-Ni-Cr Alloys

Mathematical model of selective and competitive oxidation of multi-component non ideal alloys is used for modelling oxidation of Fe-Cr-Ni alloys. The model is based on: a) the Danielewski-Holly model of interdiffusion, b) the Wagner model of the Ni-Pt alloy oxidation, c) the postulate that the values of fluxes in reacting alloy are limited (the kinetic constraint) and d) the thermodynamics of the Fe-Ni-Cr system. In this paper for the first time modelling of oxidation of a ternary non-ideal alloy based on Danielewski-Holly model is presented. The model is used to predict the evolution of component distributions in the reacting ternary Fe-Cr-Ni alloy. The results of the modelling of oxidation of the 316L stainless steel at 1173 K are presented. We compute the chromium depletion during the long term oxidation. The results allows to conclude that the oxidation reaction is limited by interdiffusion in reacting alloy. The computations demonstrate that the chromium depletion is the key factor affecting the scale stability during the long time exposition.

Modeling Non Equilibrium Potentiometry to Understand and Control Selectivity and Detection Limit

The majority of present theoretical interpretations of ion-sensor response focus on phase boundary potentials. They assume electroneutrality and equilibrium or steady-state, thus ignoring electrochemical migration and time-dependent effects, respectively. These theoretical approaches, owing to their idealizations, make theorizing on ion distributions and electrical potentials in space and time domains impossible. Moreover, they are in conflict with recent experimental reports on ion-sensors, in which both kinetic (time-dependent) discrimination of ions to improve selectivity, and non-equilibrium transmembrane ion-transport for lowering detection limits, are deliberately used. For the above reasons, the Nernst-Planck-Poisson (NPP) equations are employed here to model the non-equilibrium response in a mathematically congruent manner. In the NPP model, electroneutrality and steady-state/equilibrium assumptions are abandoned. Consequently, directly predicting and visualizing the selectivity and the low detection limit variability over time, as well as the influence of other parameters, i.e. ion diffusibility, membrane thickness and permittivity, and primary to interfering ion concentration ratios on ion-sensor responses, are possible. Additionally, the NPP allows for solving the inverse problem i.e. searching for optimal sensor properties and measurement conditions via target functions and hierarchical modeling. The conditions under which experimentally measured selectivity coefficients are true (unbiased) and detection limits are optimized are demonstrated, and practical conclusions relevant to clinical measurements and bioassays are derived.

Prediction of the Depletion Zone due to Selective Oxidation of P91 Steel

Experimental measurements do not allow for a unique determination of the concentration profiles, e.g., in case of multi-layer systems. The measured concentration of the elements at the alloy/scale interface is an average concentration in an alloy and in a scale near the spot of the beam [1]. The knowledge of the concentration of the elements at the boundary is necessary for the understanding corrosion of alloys. This essential obstacle of experimental techniques can be overcome by computer modelling. Namely, by combining the different methods (non-unique measurement with unique modelling). The Danielewski-Holly model of interdiffusion has a unique solution. This model enables to predict the evolution of component distributions in the reacting alloy. The model is valid for time dependent boundary conditions and consequently can be used for modelling the more complex reactions, eg., the formation of complex oxides. To avoid the nonphysical values of fluxes in reacting alloy the kinetic constraint on all fluxes was introduced, i.e., the flux limitation method. The results of the selective oxidation of the P91 steel (0,1 wt.% C, 8,6 wt.% Cr, 0,25 wt.% Ni) are presented. Calculated concentration profiles are compared with the experimental data. We show the evolution of chromium distribution in oxidizing steel up to 3 000 hours. The computations demonstrate that chromium depletion is the key factor determining the scale composition.

Numerical Aspects of Electrodiffusion Problem Based on Nernst-Planck and Poisson Equations

Motivation for this work comes from the application of the inverse method to electrochemical systems. The basic process operating in these systems is electrodiffusion, which can be described by the full form of the Nernst-Planck and Poisson equations. No simplification like electroneutrality assumption is used. Numerical procedure based on the method of lines (MLs) for time dependent electrodiffusion transport is presented with any number of ionic species. The resulting system of ODEs is effectively solved by employing different integrators (Radau IIA, Rosenbrock, SEULEX). Selected electrochemical systems (liquid junction, bi-ionic case, ion selective electrodes (ISE)) are treated. Performance of the integrators is compared.

Continuous Modeling of Calcium Transport Through Biological Membranes

In this work an approach to the modeling of the biological membranes where a membrane is treated as a continuous medium is presented. The Nernst-Planck-Poisson model including Poisson equation for electric potential is used to describe transport of ions in the mitochondrial membrane—the interface which joins mitochondrial matrix with cellular cytosis. The transport of calcium ions is considered. Concentration of calcium inside the mitochondrion is not known accurately because different analytical methods give dramatically different results. We explain mathematically these differences assuming the complexing reaction inside mitochondrion and the existence of the calcium set-point (concentration of calcium in cytosis below which calcium stops entering the mitochondrion).