A transfer matrix analysis of quaternary diffusion

A transfer matrix method (TMM), developed recently by Ram-Mohan and Dayananda, has been employed for the first time to generate error function solutions for quaternary diffusion in solid–solid metallic diffusion couples. The method was validated with the aid of a hypothetical couple with known quaternary interdiffusion coefficients and applied to two experimental Cu–Ni–Zn–Mn quaternary couples annealed at 775°C. Quaternary interdiffusion coefficients were determined by the Dayananda analysis from the concentration profiles of the couples over selected composition ranges in the diffusion zone and employed for the subsequent TMM calculation of error function solutions. For the hypothetical test couple, identical sets of constant interdiffusion coefficients were determined on either side of the Matano plane and utilized for the calculation of concentration profiles by TMM. For the experimental Cu–Ni–Zn–Mn couples, interdiffusion coefficients determined over selected regions in the diffusion zone were employed for the TMM generation of the concentration profiles.     

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.     

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).     

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.     

First-principles calculations for the elastic properties of Ni-base model superalloys: Ni/Ni3Al multilayers

A model system consisting of Ni[001](100)/Ni3Al[001](100) multi-layers are studied using the density functional theory in order to explore the elastic properties of single crystal Ni-based superalloys. Simulation results are consistent with the experimental observation that rafted Ni-base superalloys virtually possess a cubic symmetry. The convergence of the elastic properties with respect to the thickness of the multilayers are tested by a series of multilayers from 2γ′+2γ to 10γ′+10γ atomic layers. The elastic properties are found to vary little with the increase of the multilayer’s thickness. A Ni/Ni3Al multilayer with 10γ′+10γ atomic layers (3.54 nm) can be used to simulate the mechanical properties of Ni-base model superalloys. Our calculated elastic constants, bulk modulus, orientation-dependent shear modulus and Young’s modulus, as well as the Zener anisotropy factor are all compatible with the measured results of Ni-base model superalloys R1 and the advanced commercial superalloys TMS-26, CMSX-4 at a low temperature. The mechanical properties as a function of the γ′ phase volume fraction are calculated by varying the proportion of the γ and γ′ phase in the multilayers. Besides, the mechanical properties of two-phase Ni/Ni3Al multilayer can be well predicted by the Voigt-Reuss-Hill rule of mixtures.     

Effect of Pt on interdiffusion and mechanical properties of the γ and γ′ phases in the Ni–Pt–Al system

The effect of Pt on the growth kinetics of the γ′-[Ni(Pt)]₃Al ordered intermetallic phase and the γ-Ni(Pt, Al) solid solution diffusion rates of the species, hardness and elastic modulus was examined by employing the diffusion couple experimental technique. Experiments were conducted by using the β-Ni(Pt)Al phase and Ni(Pt) alloy couples, each of which had a fixed amount of Pt (5, 10 and 15 at. %) in both the end members so that the Pt content is more or less constant throughout the interdiffusion zone. The results suggest that the growth kinetics of both phases and the average effective interdiffusion coefficients of Ni and Al increase with the increase in Pt content. Nanoindentation studies across the compositional gradients show that the mechanical properties of the intermetallic phase in the superalloy are relatively insensitive to the presence of Pt but are more sensitive to the Ni/Al ratio. In contrast, the marked variation in the hardness of the γ phase were noted, increasing markedly with Al concentration in a given couple and also increasing with increasing Pt content. Possible causes for the observed variations are discussed.     

Interdiffusion and solid solution strengthening in Ni–Co–Pt and Ni–Co–Fe ternary systems

Diffusion couple experiments are conducted in Co–Ni–Pt system at 1200?°C and in Co–Ni–Fe system at 1150?°C, by coupling binary alloys with the third element. Uphill diffusion is observed for both Co and Ni in Pt rich corner of the Co–Ni–Pt system, whereas in the Co–Ni–Fe system, it is observed for Co. Main and cross interdiffusion coefficients are calculated at the composition of intersection of two independent diffusion profiles. In both the systems, the main interdiffusion coefficients are positive over the whole composition range and the cross interdiffusion coefficients show both positive and negative values at different regions. Hardness measured by performing the nanoindentations on diffusion couples of both the systems shows the higher values at intermediate compositions.     

Reactive diffusion in the Ti–Si system and the significance of the parabolic growth constant

Solid diffusion couple experiments are conducted to analyse the growth mechanism of the phases and the diffusion mechanism of the components in the Ti–Si system. The calculation of the parabolic growth constants and the integrated diffusion coefficients substantiates that the analysis is intrinsically prone to erroneous conclusions if it is based on just the parabolic growth constants determined for a multiphase interdiffusion zone. The location of the marker plane is detected based on the uniform grain morphology in the TiSi₂ phase, which indicates that this phase grows mainly because of Si diffusion. The growth mechanism of the phases and morphological evolution in the interdiffusion zone are explained with the help of imaginary diffusion couples. The activation enthalpies for the integrated diffusion coefficient of TiSi₂ and the Si tracer diffusion are calculated as 190 ± 9 and 197 ± 8?kJ/mol, respectively. The crystal structure, details on the nearest neighbours of the components, and their relative mobilities indicate that the vacancies are mainly present on the Si sublattice.     

Fluctuations with a 1/f α spectrum in nonequilibrium phase transitions in a spatially distributed system

Statistics of fluctuations in a spatially distributed system describing the interaction of nonequilibrium phase transitions is studied. It is shown that for a certain value of the intensity of external white noise acting on phase transitions, the time and spatial spectra of fluctuations exhibit power dependences S(f) ～ f ^?α and S(k) ～ k ^?γ. The dependence of exponents α and γ on the diffusion coefficient determining the spatial interaction of fluctuations is determined. Extremal low-frequency fluctuations are singled out and the distribution functions of their duration (P(τ) ～ τ^?β) and size (P(s) ～ s^?ν)) are constructed. It is found that exponent α in the time spectral dependence and exponent β in the duration of fluctuations are connected via the relation α + β = 2. Exponents γ and ν in the spatial spectral dependence and in the size distribution function are connected via an analogous relation (γ + ν = 2).     

稳恒磁场对Fe-Fe50 wt.%Si扩散偶中间相生长的影响

本文考察了Fe-Fe50 wt.%Si扩散偶在1200℃ 无磁场以及稳恒磁场下扩散层生长规律. 利用真空浇注强制冷却技术制备Fe-Fe50 wt.%Si扩散偶, 将制备的扩散偶进行1200℃不同磁感应强度下的热处理. 对获得热处理后试样进行SEM与EDS线扫描分析, 结果表明, 无论无磁场还是稳恒磁场下Fe-Fe50 wt.%Si扩散偶均生成两个扩散层, 即FeSi相层和Fe-Si固溶体层, 并且发现0.8 T下的两个扩散层宽度均小于0 T磁场下试样. 按照抛物线规律, 计算了扩散偶中间扩散层的互扩散系数, 发现0.8 T磁场下FeSi相层和Fe-Si固溶体层的互扩散系数较无磁场下 分别降低了26.7%与34.1%. 通过对磁吉布斯自由能的计算, 发现0.8 T磁场对扩散激活能Q的影响不足以影响扩散过程. 但扩散过程中原子振动频率ν会受到磁场的影响, 进而影响扩散常数D₀, 磁场对原子振动频率的影响可以用拉莫尔旋进理论进行解释. 关键词： Fe-Fe50wt.%Si扩散偶 稳恒磁场 FeSi相 Fe-Si固溶体     

Effects of temperature gradient on the interface microstructure and diffusion of diffusion couples:Phase-field simulation

The temporal interface microstructures and diffusions in the diffusion couples with the mutual interactions of the temperature gradient, concentration difference and initial aging time of the alloys are studied by phase-field simulation, and the diffusion couples are produced by the initial aged spinodal alloys with different compositions. Temporal composition evolution and volume fraction of the separated phase indicate the element diffusion direction through the interface under the temperature gradient. The increased temperature gradient induces a wide single-phase region on two sides of the interface.The uphill diffusion proceeds through the interface, no matter whether the diffusion direction is up or down with respect to the temperature gradient. For an alloy with short initial aging time, phase transformation accompanying the interdiffusion results in the straight interface with the single-phase regions on both sides. Compared with the temperature gradient,composition difference of diffusion couple and initial aging time of the alloy show greater effects on diffusion and interface microstructure.     

Analysis of interdiffusion via vacancy-pairs in strongly ionic solids

In this paper, we analyse chemical interdiffusion via Schottky vacancy-pairs in strongly ionic crystals for diffusion couples (AY–BY), where A and B take the same valence. We derive a sum-rule relating the phenomenological coefficients that is based on an earlier sum-rule derived by Moleko and Allnatt (Phil. Mag. A, 58 () 677) for diffusion in the multicomponent random alloy via the agency of isolated vacancies. With this sum-rule and other relationships derived by us, we show that the ratio of the intrinsic diffusivities can be expressed in exactly the same simple form as the case for diffusion via isolated vacancies. The functional form for the interdiffusion coefficient when expressed in terms of atom-vacancy exchange frequencies for diffusion is found to be essentially the same as that for isolated vacancies. The difference centres only on the relative differences of the local jump coordination of the vacancies in each case.     

Ionic diffusion on a lattice: Effects of the order-disorder transition on the dynamics of non-equilibrium systems

We examine the behaviour of the concentration profiles of particles with repulsive interactions diffusing on a host lattice. At low temperature, the diffusion process is strongly influenced by the presence of ordered domains. We use mean field equations and Monte-Carlo simulations to describe the various effects which influence the kinetic behaviour. An effective diffusion coefficient is determined analytically and is compared with the simulations. Finite gradient effects on the ordered domains and on the diffusion are discussed. The kinetics studied is relevant for superionic conductors, for intercalation and also for the diffusion of particles adsorbed on a substrate. Received: 26 June 1997 / Revised: 18 September 1997 / Accepted: 10 November 1997     

Nonadiabatic couplings and charge transfer study in H + CS+ collision using time-dependent quantum dynamics

Experiments have reported the high stability of HCS⁺ ion and inhibit to decompose over the range of collision energies. In this study, the various energy transfer channels of atomic H collision with CS⁺ molecular ion has been performed by ab initio computations at the multireference configuration interaction/aug-cc-pVQZ level of theory. The ground and several low-lying excited electronic state potential energy surfaces in three different molecular orientations, namely, two collinear configurations with, (1) H approaching the S atom (γ = 0°), (2) H approaching the C atom (γ = 180°) and one perpendicular configuration, (3) H approaching the centre of mass of CS (γ = 90°) with the diatom fixed at the equilibrium bond length, have been obtained. Nonadiabatic effects with Landau–Zener coupling leading to avoided crossings are observed between the ground- and the first-excited states in γ = 90° orientation, and also between the first- and second-excited states in γ = 180° orientation. Quantum dynamics have been performed to study the charge transfer using time-dependent wave packet method on the diabatic potential energy surfaces. The probability of charge transfer is found to be highest with 42% in γ = 180°. The high charge transfer probability result in the formation of H⁺ + CS channel which ascertains the high stability of HCS⁺ ion.     

The Kirkendall effect in single-phase multicomponent systems: dependence on drift and entropy distribution

This paper concerns interdiffusion in a diffusion couple and determination of the Kirkendall plane. The “entropy density” model is proposed in which the entropy is used to predict the position of the Kirkendall plane in a multicomponent system. It is shown that the marker position depends on the drift velocity and pressure field only. Application of the model is presented for ternary CoFeNi diffusion couples of three various initial compositions. The concentration profiles and entropy densities are calculated for each diffusion couple. The positions of the Kirkenadl planes are determined and compared with those obtained by velocity-curve and trajectory methods.     

Diffraction profiles of thin film diffusion couples

On the basis of hypothetical thin film couples in which a grain-boundary supported diffusion process is simulated, it is shown that the concentration spectrum may contain maxima which would not occur if the process were governed by lattice diffusion only. As such maxima consequently appear in the diffraction pattern, real diffraction profiles of interdiffusion layers must be interpreted carefully, keeping in mind that diffraction peaks need not necessarily be associated with a particular phase. The thin-film diffusion system Au-Cu is chosen as an example to demonstrate experimentally the generation of diffraction maxima which must not be attributed to phase formation.Dedicated to Prof. Dr. F. Lihl on the occasion of his 75^th birthday     

Electron capture and loss in electron swarm experiments

The effect of capture and subsequent loss of electrons upon their time distribution in time-of-flight electron swarm experiments has been investigated. It is found that if the mean release time of captured electrons is much smaller than both the drift time and the time width of the electron swarm distribution, the effective diffusion coefficientD′ is given approximately by whereD is the electron diffusion coefficient in the absence of capture,w is the drift velocity, γ is the inverse release time, β is the inverse capture time and ζ=ψ/β+γ It is shown that measurements ofw andD by the steady state Townsend method should be unaffected by temporary capture processes.     

Diffusion interaction between Al and Mg controlled by a high magnetic field

The effects of an 11.5 T high magnetic field on the growth behavior of compounds layers during reactive diffusion in solid Al/solid Mg diffusion couples have been investigated. After annealing with and without a magnetic field, the interfacial zone was still composed of two layers of stable compounds β (Mg₂Al₃) and γ (Mg₁₇Al₁₂), but the thicknesses of the layers were increased by the magnetic field. Data analysis of the thicknesses of compound layers showed that the growth of the γ layer was controlled by grain boundary diffusion as well as volume diffusion at B=0 T, but only controlled by volume diffusion when an 11.5 T high magnetic field was applied. The interdiffusion coefficients for the samples annealed at 643–693 K were calculated from the parabolic relationship between the migration of the interface and the annealing time. The results show that the diffusion constant increased due to the high magnetic field, increasing the chemical potential gradient, and in turn it caused the interdiffusion coefficient to increase.     

Electrochemical study of the silver-sodium substitution in a borosilicate glass

The study of the silver for sodium ions substitution kinetics is carried out using an electrochemical set-up. A silver metal anode, a reference and counter electrode are deposited as thin films on a 0.6 SiO₂-0.2 B₂O₃-0.2 Na₂O glassy electrolyte. The anode dissolution is studied by a chronoamperometric method. The generally proposed interdiffusion models assuming constant diffusion coefficients for both Ag⁺ and Na⁺ cations are not able to justify the experimental voltage time dependence. The deviation from theoretical predictions can be minimised by assuming a mixed cation effect in the glass, taking into account a concentration dependence of the Ag⁺ and Na⁺ diffusion coefficients.