Statistical theory of diffusion in concentrated bcc and fcc alloys and concentration dependencies of diffusion coefficients in bcc alloys FeCu,FeMn, FeNi,and FeCr

The statistical theory of diffusion in concentrated bcc and fcc alloys with arbitrary pairwise interatomic interactions based on the master equation approach is developed. Vacancy–atom correlations are described using both the second-shell-jump and the nearest-neighbor-jump approximations which are shown to be usually sufficiently accurate. General expressions for Onsager coefficients in terms of microscopic interatomic interactions and some statistical averages are given. Both the analytical kinetic mean-field and the Monte Carlo methods for finding these averages are described. The theory developed is used to describe sharp concentration dependencies of diffusion coefficients in several iron-based alloy systems. For the bcc alloys FeCu, FeMn, and FeNi, we predict the notable increase of the iron self-diffusion coefficient with solute concentration c, up to several times, even though values of c possible for these alloys do not exceed some percent. For the bcc alloys FeCr at high temperatures T ? 1400 K, we show that the very strong and peculiar concentration dependencies of both tracer and chemical diffusion coefficients observed in these alloys can be naturally explained by the theory, without invoking exotic models discussed earlier.     

Surface diffusion of particles adsorbed on an inhomogeneous lattice with two non-equivalent sites

We investigate surface diffusion in a system of particles with the nearest neighbor pairwise lateral interaction adsorbed on a two-dimensional inhomogeneous lattice of square symmetry with deep and shallow sites. General analytical expressions for the chemical and jump diffusion coefficients have been derived in case of strong inhomogeneity. The expressions are valid for the inhomogeneous lattices with different geometries and dimensionalities. We have calculated the coverage dependencies of the tracer, jump and chemical diffusion coefficients for different temperatures using the real-space renormalization group (RSRG) method and compared the data with the numerical results obtained by the MC simulations. The coincidence between the data obtained by these quite different methods is rather good.     

Simultaneous tracer diffusion and interdiffusion in a sandwich-type configuration to provide the composition dependence of the tracer diffusion coefficients

In this paper, a new formalism of a combined tracer and interdiffusion experiment for a binary interdiffusion couple is developed. The analysis requires an interdiffusion couple that initially contains a thin layer of tracers of one or both of the constituent elements at the original interface of the couple (sandwich interdiffusion experiment). This type of interdiffusion experiment was first performed in 1958 by J.R. Manning. The theoretical analysis presented in this paper is based on a newly developed phenomenological theory of isotopic interdiffusion combined with the Boltzmann–Matano formalism. This new analysis now provides the means to obtain the composition dependent interdiffusion coefficient and tracer diffusion coefficients simultaneously from analysis of the interdiffusion and tracer profiles in a single sandwich interdiffusion experiment. The new analysis is successfully applied to the results of Manning’s original ‘sandwich interdiffusion’ experiment in the Ag–Cd system (six couples in total) and is validated with an independent determination of the Ag and Cd tracer diffusion coefficients by Schoen using the conventional thin film technique. Suggestions for further development of the sandwich interdiffusion experiment and analysis to the case of multicomponent alloys are provided.     

Contribution a la theorie atomique de la diffusion chimique

Here we present an analysis and a development of the atomic theory of chemical diffusion as proposed by Manning for a binary system a/b.The general expression for the flux of a tracer in a concentration gradient is first established. This expression of the flux is identified with that deduced in the phenomenological theory. Thus a relationship between the partial correlation factors of vacancies with each of the a and b species is obtained.The effect of “vacancy flow” can be described in terms of these correlation factors. Thus the vacancy flow on species A leads to a correlation of the vacancy jumps with species B and vice versa.We shall see that the Nernst-Einstein equation can be extended to the case of chemical diffusion and that the ratio of the intrinsic diffusion coefficients is equal to the ratio of the mean jump frequencies W_A and W_B.Also, the activation energies of intrinsic diffusion coefficients are related very simply to the activation enthalpies of atomic jumps.In conclusion, we shall see that chemical diffusion in a binary system a/b can be completely described if either the thermodynamic factor and the coefficients of self diffusion, or the thermodynamic factor and the coefficients of intrinsic diffusion are known as functions of the concentration.     

Random alloy diffusion kinetics for the application to multicomponent alloy systems

In this paper, extensive Monte Carlo simulation results are reported on tracer and collective diffusion correlation effects in the random ternary alloy, as an example of a multicomponent alloy system. The problem of analytically describing both collective and tracer diffusion kinetics is also addressed for the random multicomponent alloy by application of a combination of the Manning theory and Holdsworth and Elliott theory. It is found that the overall results from the combined theory agree reasonably well with Monte Carlo results. This combined approach is much more accurate than Manning’s approach itself and much more manageable than the almost exact, but unfortunately difficult to use, self-consistent theory of Moleko, Allnatt and Allnatt. Some relations between the Onsager phenomenological coefficients and tracer diffusion coefficients are derived and are tested with our Monte Carlo data. Good agreement is found.     

The Manning factor for direct exchange and ring diffusion mechanisms

Abstract

In this paper, we consider lattice-based diffusion kinetics for the direct exchange and ring mechanisms as possible proxy diffusion mechanisms for diffusion in liquid alloys. For these mechanisms, we assessed the Manning factor that arises from the Darken–Manning relation relating the interdiffusion coefficient and tracer diffusion coefficients and which can be obtained experimentally. The maximum values of the Manning factor for these two mechanisms occur when the exchange only takes place between the atoms of different type but not between the atoms of the same type. These values have strong composition dependence and reach a value of 2 (ignoring tracer correlation factors) for the direct exchange mechanism at equal compositions of the two components in binary alloys. But for the three atom ring mechanism, these values as a function of composition have a much more complicated form that sits below the direct exchange mechanism for compositions between 10 and 90%. When all exchanges (allowed by a mechanism) occur with approximately the same probability, then the Manning factor is about unity for all compositions.     

Diffusion of particles on a heterogeneous surface: A case of adsorption-induced surface reconstruction

The influence of surface reconstruction on diffusion of particles adsorbed on the surface is investigated in the framework of symmetrical four-position model. The analytical expressions for free energy and diffusion coefficients are obtained assuming the lateral interaction between particles is negligibly small.The critical behavior of the system is described by the Ising spin model. The coverage dependencies of the tracer, jump and chemical diffusion coefficients are calculated for some representative temperatures. The dependencies show clearly strong influence of the surface reconstruction on the thermodynamic and kinetic phenomena: diffusion coefficients become anisotropic on the reconstructed surface. To check the analytical results we have used Monte Carlo simulations of the diffusion on this lattice.     

Pseudo-binary and pseudo-ternary diffusion couple methods for estimation of the diffusion coefficients in multicomponent systems and high entropy alloys

The benefits of using the pseudo-binary and pseudo-ternary diffusion couple methods in multicomponent inhomogeneous systems are demonstrated by estimating different types of composition-dependent diffusion coefficients. These are important for understanding the basic atomic mechanism of diffusion and complex compositional evolutions. These were otherwise considered impossible during the last many decades. Without any options previously, sometimes the average values over a composition range of random choice were estimated, which are not the material constants but depend on the composition range and also the end member compositions. The steps and analyses for utilising the pseudo-binary and pseudo-ternary methods are first described in the Ni-Co-Fe-Mo system by producing the ideal diffusion profiles fulfilling the concepts behind these methods. Following, the discussion is extended to the systems related to medium (Ni-Co-Cr) and high (Ni-Co-Fe-Mn-Al) entropy alloys. In fact, this is the first report showing a correct experimental method that should be followed for the estimation of the interdiffusion and intrinsic diffusion coefficients in inhomogeneous high entropy alloys. In the end, the limitations of following these methods because of the generation of non-ideal diffusion profiles are discussed based on experimental results. The steps are also suggested to avoid such complications. These methods are easy to adopt for research engineers. Most importantly, these give an opportunity to validate the data estimated following newly proposed numerical methods by different groups with experimentally estimated diffusion coefficients, which were not possible earlier.     

Chapman-Enskog velocity distribution for tracer diffusion

The Chapman-Enskog method is used to obtain an approximate velocity distribution function for tracer diffusion in dilute hard-sphere mixtures. Different ratios of the mass of the tracer to that of the excess component (including the well-known limiting cases of the Lorentz and the Rayleigh models) are considered and the corresponding diffusion coefficients are also evaluated. A comparison with the recent results of Tompson and Loyalka for both the diffusion coefficients and the distribution functions provides a perspective on the usefulness and nature of the approximate method.     

Diffusion of particles over strongly anisotropic surface with traps

We investigate surface diffusion in a system of particles adsorbed on a two-dimensional strongly anisotropic lattice. There are two kinds of the lattice sites - ordinary sites and deep traps. Particles adsorbed in the ordinary sites can migrate over the surface, but particles adsorbed in traps are immobile. These particles do not move over the surface and they obstacle also the mobile particles migration (surface defects). Using kinetic Monte Carlo simulations we obtained coverage dependencies of the tracer, jump, and chemical diffusion coefficients. The coefficients are rather sensitive to the defect concentration. Even small admixture of the defects decreases drastically the fast diffusion. The effect is rather specific: strong dependence of the pre-exponential factor on the defect concentration and almost independent activation energy. The defect influence on the slow diffusion is weak. It results in strong decreasing of the surface diffusion anisotropy with the defect concentration. Such unusual behavior of the diffusion coefficients was observed in many experimental investigations of the surface diffusion of lithium, cesium, potassium, and strontium over strongly anisotropic W(1 1 2) and Mo(1 1 2) planes. It was shown that this specific behavior arises exclusively due to the surface anisotropy, and does not depend on the lateral interaction between the particles.     

Implantation and diffusion of 73As in GaAs and GaP

Self-diffusion on the As sublattice in intrinsic GaAs and foreign-atom diffusion on the P sublattice in intrinsic GaP were investigated in a direct way by As tracer diffusion measurements using the radioisotope ⁷³As. For this purpose ⁷³As was implanted in both materials at the ISOLDE facility of CERN. Then diffusion annealings were performed followed by serial sectioning and counting of the radioactivity in each section. The resulting profiles were simulated within a computer model which accounts for the observed loss of tracer to the diffusion ambient. The so-obtained diffusion coefficients for As in GaAs and GaP are compared with existing diffusivities in these compounds. This revised version was published online in July 2006 with corrections to the Cover Date.     

Surface tracer diffusion in the presence of a solid barrier: a Monte Carlo study

We investigate, by way of Monte Carlo simulations, the diffusion of adsorbates on a square lattice. The lattice contains inaccessible sites forming linear barriers. The adsorbate atoms do not interact among themselves, except for the hard core. The tracer diffusion coefficients, reduced by their values in the absence of the barrier, are plotted versus the length of the barrier. We find that the diffusion along the barrier is not affected by its presence, whereas that perpendicular to it decreases linearly, for a large interval of the barrier, with its length. The reduced diffusion coefficient does not depend on the concentration of the diffusing particles, but its value increases with the perpendicular distance between the barriers.     

Influence of isotopic substitution on the diffusion and thermal diffusion coefficient of binary liquids

The mutual mass diffusion coefficient (D) and the thermal diffusion coefficient ( D _T) of the liquids acetone, benzene, benzene-d ₁, benzene-d ₃, benzene-d ₅, benzene-d ₆, benzene- ¹³C₆, n-hexane, toluene, 1, 2, 3, 4-tetrahydronaphtalene, isobutylbenzene, and 1, 6-dibromohexane in protonated and perdeuterated cyclohexane have been measured with a transient holographic grating technique at a temperature of 25 ^°C. The mass diffusion coefficient shows a pronounced concentration dependence. Perdeuteration of cyclohexane only leads to marginal changes of the mass diffusion coefficient. The Stokes-Einstein equation describes the limiting tracer diffusion coefficients well if the solute molecule is smaller than the solvent. It is not capable to describe the small isotope effect of a few percent. On the other hand, the isotope effect, which is independent of concentration, is in agreement with the Enskog theory, that does not provide the absolute value of the mass diffusion coefficient of the liquid mixtures. The thermal diffusion coefficient of all the binary mixtures shows a moderate and almost linear concentration dependence. Its isotope effect, which is the change of D _T upon deuteration of cyclohexane, varies with mole fraction. The thermophoretic force acting on any tracer molecule in cyclohexane changes by the same amount when cyclohexane is perdeuterated, irrespective of the magnitude of the thermophoretic force before deuteration. This change of the thermophoretic force is equal but of opposite sign to the difference between the thermophoretic forces acting on cyclohexane and perdeuterated cyclohexane as tracers in any of the above liquids.     

A dynamical mean field theory for the study of surface diffusion constants

We present a combined analytical and numerical approach based on the Mori projection operator formalism and Monte Carlo simulations to study surface diffusion within the lattice-gas model. In the present theory, the average jump rate and the susceptibility factor appearing are evaluated through Monte Carlo simulations, while the memory functions are approximated by the known results for a Langmuir gas model. This leads to a dynamical mean field theory (DMF) for collective diffusion, while approximate correlation effects beyond DMF are included for tracer diffusion. We apply our formalism to three very different strongly interacting systems and compare the results of the new approach with those of usual Monte Carlo simulations. We find that the combined approach works very well for collective diffusion, whereas for tracer diffusion the influence of interactions on the memory effects is more prominent.     

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.     

Unified analysis of diffusion and relaxation processes in amorphous metallic alloys

Abstract

Self-diffusion data on amorphous metallic alloys determined in long-time radio-tracer experiments and activation-enthalpy spectra deduced from short-time structural relaxation studies on such materials are reviewed and analyzed in terms of current random transition rate models. It is shown that the seeming discrepancy between the self-diffusion enthalpies and the comparatively small enthalpies obtained from measurements of the magnetic after-effect arises from the existence of activation-enthalpy spectra. Whereas the short-time experiments reveal the small-activation-enthalpy parts of these spectra, the long-time self-diffusion experiments are controlled by the larger activation enthalpies. Assuming that the activation-enthalpy distributions are Gaussian, their characteristic parameters have been determined by comparing the two types of measurements. It is shown that the Arrhenius-type diffusion coefficients found by experiment are compatible with half-widths of the activation-enthalpy spectra of about 0.3 eV. Based on an analysis of the pre-exponential factors and other typical properties of the diffusion coefficients, potential mechanisms of the diffusion in amorphous alloys are proposed.     

Simulation of retarded diffusion of antimony and enhanced diffusion of phosphorus in silicon

The paper is concerned with the influence of point defects on dopant diffusion in silicon. This influence is analysed by means of comparing the experimental results of Mizuo and Higuchi with our simulation. The experiments deal with the oxidation retarded diffusion (ORD) of Sb and the oxidation enhanced diffusion (OED) ofP in FZ silicon at 1100°C. The simulation is carried out by using a physical model without simplifying assumptions. It is shown that simplifying assumptions are not admissible. A reasonable set of parameters is deduced from this analysis. As each parameter represents a physical effect, information about the importance of bulk and surface recombination of point defects and about the equilibrium concentration values and diffusion coefficients of diffusing species are obtained. It turns out that the influence of the surface is decisive for the distribution of point defects.Dedicated to Professor Karlheinz Seeger on the occasion of his 60th birthday