Reaction diffusion and interdiffusion in some binary metallic systems

This paper reviews studies from the Münster laboratory on multiphase and interdiffusion performed on several binary metallic systems. Optical microscopy and electron microprobe analysis were used to study interdiffusion and growth of the intermetallic layer(s) in infinite diffusion couples. Both methods have resolutions at the micron level. Thus, the results summarised in this paper concern the diffusion-controlled regime of the reaction diffusion process. We first remind the reader of some of the basics of multiphase diffusion and interdiffusion. Then we discuss results reported for the following systems: cobalt–niobium, nickel–niobium, nickel–aluminium and magnesium–aluminium. In the case of Co₂Nb we also compare interdiffusion and tracer diffusion of the components. Multiphase diffusion experiments also contain information about solid–solid equilibria of the phase diagram(s).     

Accurate measurement of translational diffusion coefficients: a practical method to account for nonlinear gradients

For NMR probes equipped with pulsed field gradient coils, which are not optimized for gradient linearity, the precision and accuracy of experimentally measured translational diffusion coefficients are limited by the linearity of the gradient pulses over the sample volume. This study shows that the accuracy and precision of measured diffusion coefficients by the Stejskal--Tanner spin-echo pulsed field gradient experiment can be significantly improved by mapping the gradient z-profile and by using the mapped calibration parameters in the data analysis. For practical applications the gradient distribution may be approximated by a truncated linear distribution defined by minimum and maximum values of the gradient. By including the truncated linear gradient distribution function in the Stejskal--Tanner equation, the systematic deviation between the fitted curve and the experimental attenuation curve decreases by an order of magnitude. The gradient distribution may be calibrated using an intense NMR signal from a sample with a known diffusion coefficient. The diffusion coefficient of an unknown sample may then be determined from a two-parameter fit, using the known gradient distribution function.     

Analysis of composition profiles to obtain interdiffusion coefficients in binary intermetallic compounds

Intermetallic compounds containing high concentrations of structural vacancies have made the determination of the interdiffusivity from the composition profile from a diffusion couple a confusing issue. In this paper, an exact procedure is formulated from fundamental principles. The only assumption made is the use of the Gibbs–Duhem relation. We show that the standard procedure for analysing composition profiles in diffusion couples slightly overestimates the interdiffusivity. We also show that the procedure put forward by Kim and Chang (Metall. Mater. Trans., 31A () 1519) to analyse the composition profiles in NiAl is incorrect near the stoichiometric composition where it seriously underestimates the interdiffusivity.     

Statistical calculations of tracer and intrinsic diffusion coefficients in concentrated alloys and estimates of microscopic parameters of diffusion from experimental data

The earlier-developed statistical theory of diffusion in concentrated alloys based on the master equation approach is generalized to treat tracer diffusion in binary alloys. The theory developed is used to describe concentration dependencies of both tracer and intrinsic diffusion coefficients and to estimate microscopic parameters of diffusion in alloys CuNi, CuZn and AgCd for which necessary experimental data are available. We show that all main features of strong and peculiar concentration dependencies of diffusion coefficients observed in these alloys are naturally explained by the theory. Signs and scales of interatomic interactions important for diffusion in these alloys are found to strongly depend on the ratio of atomic sizes of alloy components, and types of these dependencies agree with simple physical considerations. We also discuss physical reasons for sharp concentration dependencies of diffusion coefficients characteristic of real alloys.     

Grain boundary interdiffusion in the case of concentration-dependent grain boundary diffusion coefficient

The grain boundary diffusion in a binary system which exhibits a grain boundary phase transition is considered in the framework of Fisher's model. The kinetic law of the growth of the grain boundary phase and the distribution of the diffusant near the grain boundary are calculated. The method of determining of the concentration dependence of the grain boundary diffusion coefficient from the experimentally measured penetration profiles of the diffusant along the grain boundaries is suggested. The experimental results on Zn diffusion in Fe(Si) bicrystals, Ni diffusion in Cu bicrystals and grain boundary grooving in Al in the presence of liquid In are discussed in light of the suggested model.     

Reptation and interdiffusion in polystyrene networks

We have used helium-3 nuclear reaction analysis to measure the interdiffusion of linear polystyrene into a film of crosslinked polystyrene and the intradiffusion of polystyrene in polystyrene networks. The interdiffusion is compared with that predicted from the Kramer-Sillescu theory, and is found to be considerably faster. This is attributed to the relaxation of inhomogeneities in the network. The molecular weight and crosslinking dependence of the intradiffusion coefficients of free polystyrene chains trapped inside networks is discussed in terms of the simple tube model and provides reasonably good agreement with that predicted from reptation theory. Received 2 August 2000 and Received in final form 6 November 2000     

Composition dependence of the ionic diffusion coefficients in yttria-stabilized zirconias

The diffusion coefficient of the O²⁻ ions in yttria-stabilized zirconia (1−x)ZrO₂·xYO_1.5 has been investigated as a function of the concentration of Y³⁺ ion over a range x = 0.131–0.261 by using quasielastic light scattering. The diffusion coefficients were deduced from the analysis of the temperature dependence of the scattering intensity at 3 GHz. It is concluded that the diffusion coefficient reaches its maximum value at x = 0.165. The composition dependence of the scattering intensity suggests that the concentration of the moving carrier decreases as the Y³⁺ content increases in spite of an increase of the oxygen vacancies.     

Phase-field simulation of the effect of interaction among ordered domains on interdiffusion in Ni--Al--Cr alloys

The effect of interaction among γ ' ordered domains on the interdiffusion process in γ +γ ' /γ and γ +γ ' /γ +γ ' diffusion couples is investigated by using the phase-field method, in which bulk free energy and mobility are linked with thermodynamic and kinetic databases. Simulated results show that the interaction among γ ' ordered domains has great influence on the microstructure, the interdiffusion velocity and the volume fraction of γ ' phase on both sides of the diffusion couples.     

A simplified method to determine the coverage dependence of surface diffusion coefficients

A new method to determine the coverage dependence of surface diffusion coefficients from coverage-gradient experiments is presented. The initial coverage-gradient employed is an evacuated circular region surrounded by a constant coverage. In order to demonstrate this method, Fick's second law equation was numerically solved using coverage-dependent surface diffusion coefficients. An analysis of the results demonstrates that the coverage dependence of the surface diffusion coefficient can be determined using a simple extrapolation procedure. The coverage dependence can be obtained for a wide range of functional forms for the surface diffusion coefficient. The extrapolation procedure requires only that the number of adsorbates that have diffused back into the initially evacuated region be measured as a function of time. This method should be useful in the interpretation of coverage-gradient surface diffusion experiments, especially those performed using laser-induced thermal desorption techniques.     

Influence of a uniform driving force on tracer diffusion in a one-dimensional hard-core lattice gas

The influence of a uniform driving force on tracer diffusion is investigated for a one-dimensional lattice gas where particles jump stochastically to unoccupied neighboring sites. A new, simple calculation is presented for the diffusion coefficient of a tracer particle with respect to its average drift, obtained recently by rigorous methods by De Masi and Ferrari. A theoretical expression describing the tracer particle mean square displacement approximately for all times is derived and found to be in excellent agreement with the results of Monte Carlo simulations.     

On the formation of amorphous layers by interdiffusion in zirconium-copper system

Formation of amorphous phase by solid state reaction has been observed in bulk copper-zirconium couples. Transmission electron microscopy (TEM), Auger electron spectroscopy (AES) and differential scanning calorimetry (DSC) techniques have been employed to detect and study the amorphous phase. The experimental observations have been explained with a model, which considers the breakdown of the a-zirconium lattice due to the solute enrichment beyond a certain limit where the nucleation of the amorphous phase becomes thermodynamically possible. The present-experiments convincingly show that amorphous phase can form in bulk diffusion couples made from large-grained well-annealed material and conclusively establish that the presence of crystal defects in high concentrations is not a pre-requisite for the formation of amorphous phase.     

Application of a radioactive tracer method for diffusion study in some liquids

In this paper a radioactive tracer technique based on sliding cell method developed in our laboratory for the study of diffusion in liquids is presented in detail. This method consists of radioactive and non-radioactive liquid columns of equal length and the radiation detector is placed in a vertical geometry over the diffusion column. Some liquid metals and aqueous electrolyte solutions were studied by this method. The data of liquid metals like mercury and gallium have been analyzed from the point of view of hard sphere model and those of electrolyte solutions from phenomenological theories. Onsager’s coefficientsL ₁₁,L ₁₂,L ₂₂ have been calculated from the experimental data and the variation of diffusion with solute concentration has been explained from ion-ion interaction.     

A pseudobinary approach to study interdiffusion and the Kirkendall effect in multicomponent systems

Interdiffusion studies become increasingly difficult to perform with the increasing number of elements in a system. It is rather easy to calculate the interdiffusion coefficients for all the compositions in the interdiffusion zone in a binary system. The intrinsic diffusion coefficients can be calculated for the composition of Kirkendall marker plane in a binary system. In a ternary system, however, the interdiffusion coefficients can only be calculated for the composition where composition profiles from two different diffusion couples intersect. Intrinsic diffusion coefficients are possible to calculate when the Kirkendall markers are also present at that composition, which is a condition that is generally difficult to satisfy. In a quaternary system, the composition profiles for three different diffusion couples must intersect at one particular composition to calculate the diffusion parameters, which is a condition that is almost impossible to satisfy. To avoid these complications in a multicomponent system, the average interdiffusion coefficients are calculated. I propose a method of calculating the intrinsic diffusion coefficients and the variation in the interdiffusion coefficients for multicomponent systems. This method can be used for a single diffusion couple in a multicomponent pseudobinary system. The compositions of the end members of a diffusion couple should be selected such that only two elements diffuse into the interdiffusion zone. A few hypothetical diffusion couples are considered in order to validate and explain our method. Various sources of error in the calculations are also discussed.     

First principles calculations of alloying element diffusion coefficients in Ni using the five-frequency model

The diffusion coefficients of several alloying elements(Al,Mo,Co,Ta,Ru,W,Cr,Re) in Ni are directly calculated using the five-frequency model and the first principles density functional theory.The correlation factors provided by the five-frequency model are explicitly calculated.The calculated diffusion coefficients show their excellent agreement with the available experimental data.Both the diffusion pre-factor(D 0) and the activation energy(Q) of impurity diffusion are obtained.The diffusion coefficients above 700 K are sorted in the following order:DAl>DCr>DCo>DTa>DMo>DRu>DW>D Re.It is found that there is a positive correlation between the atomic radius of the solute and the jump energy of Ni that results in the rotation of the solute-vacancy pair(E 1).The value of E 2-E 1(E 2 is the solute diffusion energy) and the correlation factor each also show a positive correlation.The larger atoms in the same series have lower diffusion activation energies and faster diffusion coefficients.     

Effect of interdiffusion of In and Al atoms on excitonic states in In<Subscript>x</Subscript>Ga<Subscript>1−x</Subscript>As/Al<Subscript>y</Subscript>Ga<Subscript>1−y</Subscript>As quantum dots

The effect of interdiffusion of aluminum and indium atoms on the exciton emission energy and binding energy in In_xGa_1?xAs/Al_yGa_1?yAs quantum dots is studied. It is shown that the emission energy increases monotonically with increasing diffusion length, while the binding energy has a maximum.     

Sample-shape dependence of the spinodals and the diffusion coefficients for hydrogen in niobium and tantalum

The spinodals for H in Nb and Ta are measured for wire- and foil-shaped specimens, using the Gorsky effect. A pronounced dependence on sample geometry is found and explained in the scope of the mode theory by Wagner and Horner. Also a shape dependence of the macroscopic diffusion coefficient as predicted by this theory has experimentally been verified.     

The relationship between chemical and tracer diffusion coefficients of aliovalent ions in an ionic lattice

The theoretical model developed by Lidiard was extended to describe the relationship between the chemical and tracer diffusion coefficients of aliovalent ions in an ionic lattice.It is shown that the relationship between the chemical diffusion coefficient, D, and the tracer diffusion coefficient, D¹, is D = 2D¹ if the migration of dimers is the principal mechanism of transport and for the migration of trimers D = 3D¹ if the concentration of impurity ion is relatively small. These relationships are valid regardless of the charge of the aliovalent or lattice ions.The chemical diffusion coefficients of Cr³⁺ in Cr-doped MgO were determined for three different temperatures, 1656, 1717 and 1768K, and for the concentration region 2.5×10^?2?2.8×10^?1 mole% Cr₂O₃. Using previously determined values for the tracer diffusion coefficient of ⁵¹Cr in Cr-doped MgO it was found that for the temperature and concentration region investigated D = (2.00±0.17)D¹ which indicates that diffusion proceeds primarily by the migration of dimers.     

Effect of In and Al interdiffusion on electron states and light absorption in InxGa1? x As/AlyGa1? y As quantum dots

A new method of theoretical investigation of the interdiffusion effect on electron states in quantum dots is proposed. The main point of the method is the replacement of the “veritable” confining potential formed due to the diffusion by a model potential, for which the Schr?dinger equation solutions and the energy spectrum are known. In the framework of the proposed method we calculate the positions of edges of the conduction and heavy hole bands and the absorption coefficient of interband transitions depending on the diffusion length in spherical In_xGa_1−x As/Al_yGa_1−y As quantum dots, using the Wood-Saxon potential as a model one. Original Russian Text ? V.N. Mughnetsyan, A.A. Kirakosyan, 2007, published in Izvestiya NAN Armenii, Fizika, 2007, Vol. 42, No. 2, pp. 83–91.     

The influence of surface forces on shear-induced tracer diffusion in mono and bidisperse suspensions

The shear-induced self-diffusivity of tracer particles of radius a _t = λa in a suspension of particles having a radius, a , is calculated by Stokesian dynamics for different values of the size ratio, λ , both in 2 and 3 dimensions in the binary-collision regime. The self-diffusion is found to decrease strongly when the size ratio becomes quite different from unity. On the other hand, for the same average distance of contact between two spheres, the presence of a soft force always increases greatly the diffusion compared to the effect of a hard shell which is used to model the roughness. This is particularly true for tracer particles smaller than the bath particles, where the shear-induced diffusion can be increased by many order of magnitudes in the presence of a soft force. For suspensions of monodisperse particles we show that, for low volume fraction, the diffusion coefficient is much smaller than the one predicted by the binary collision model, due to the existence of a layered structure. On the contrary at higher volume fraction, many-body collisions strongly enhance the diffusion and it appears that the value of the diffusion is quite sensitive to the presence of clusters of particles which are themselves determined by the range of interparticle forces.