Diffusion simulation based on atomistic model and mobility determination via Kirkendall experiment for multi-component systems

Study of diffusion and atomic mobilities for fcc Ag–Cd and Ag–Sn solder alloys

Diffusion in the Ag-Cd system was investigated at 600°C with vapor-solid and solid-solid diffusion couples in the α-phase for the determination of intrinsic diffusion coefficients, atomic mobilities and interdiffus ion coefficients. Intrinsic diffusion coefficients, DCd and DAg, determined experimentally, were compared with those predicted from the models of Darken, Manning, and Dayananda with the aid of tracer diffusion and thermodynamic data. The experimental coefficients for various alpha alloys were found to be higher than those evaluated from each of the three models. On the other hand, the experimental values of the ratio, DCd/DAg were found to be in good agreement with the models of Manning and Dayananda where the vacancy wind effects on intrinsic diffusion are taken into account. Darken’s relations are considered inadequate to interrelate tracer and intrinsic diffusion coefficients in the Ag-Cd system, since vacancy wind effects are found to be appreciable in this system.

Atoms and molecules are not completely immobile within a solid material. They move by jumping into vacancies or interstitial sites in the crystal lattice. The laws describing their motion were discovered by Adolf Fick in the mid-19th century, modelled on analogous laws for the flow of heat (Fourier’s law) and electricity (Ohm’s law). According to Fick’s first law, the rate at which atoms move is proportional to the concentration gradient, with the diffusion coefficient defined as the constant of proportionality. Fick’s second law generalises the first law to a wide range of situations and is called the diffusion equation. This chapter examines a number of characteristic diffusion profiles; the difference between self, intrinsic, inter- and tracer diffusion coefficients; the Kirkendall effect and porosity formation when different components move at different speeds; and the Arrhenius temperature dependence of diffusion. Fick was a physiologist and derived his laws initially to describe the flow of blood through the heart. He made advances in anatomy, physiology and medicine, developing methods of monitoring blood pressure, muscular power, corneal pressure and glaucoma. He lived at the time of Bismarck’s post-Napoléonic unification of Germany and the associated flowering of German science, engineering, medicine and culture.

Interdiffusion coefficients in copper-rich copper-antimony solid solutions containing up to 5 at% of Sb have been determined by Matano’s method in the temperature range between 919 and 1040 K using semi-infinite diffusion couples consisting of pure copper and Cu–Sb alloys with the Kirkendall markers. From the interdiffusion coefficient and the marker shift, the intrinsic diffusion coefficients have been determined at the marker positions, 1.0 and 1.7 at%Sb. Using thin plate couples, the intrinsic diffusion coefficients in Cu-0.6 at%Sb alloy at 1005 K have been determined by Heumann’s method. From the value of the intrinsic diffusion coefficient of copper in the infinite dilution of Sb and the tracer diffusion coefficients of Sb and Cu in pure Cu, the vacancy flow factor for the interdiffusion in Cu–Sb alloys at 1005 K has been calculated to be −1.20−0.28+0.22, which suggests a strong interaction between an antimony atom and a vacancy. Combining the present result on the vacancy flow factor with the previous experimental data on the enhancement factor of solvent diffusivity by the addition of solute atoms, the correlation factor and the jump frequency ratios in the five-frequency model have been estimated. The correlation factor for the impurity diffusion of Sb in Cu thus obtained is about 0.1 at 1005 K, which indicates highly correlated jumps of an antimony atom to a vacancy.

Correlation between estimated diffusion coefficients from different types of diffusion couples in multicomponent system

The effect of large vacancy concentration on intrinsic and interdiffusion coefficients: A first-principle study of B2-NiAl

Prediction of the mutual diffusivity in binary liquid mixtures containing one dimerising species, from the tracer diffusion coefficients

A robust and intelligent computational framework for assessing diffusion coefficients in alloys

Atomic mobility, diffusivity and diffusion growth simulation for fcc Cu–Mn–Ni alloys

We present new experimental data on diffusion of divalent cations in almandine-spessartine diffusion couples in graphite capsules in the P-T range of 14–35 kb, 1100–1200° C. The tracer diffusion coefficients of the major divalent cations, viz. Fe, Mg and Mn, retrieved from the multicomponent diffusion profiles, have been combined with earlier data from our laboratory at 29–43 kb, 1300–1480° C (Loomis et al. 1985) to derive expressions of the P-T dependence of the diffusion coefficients at fO2 approximately corresponding to that defined by equilibrium in the system graphite-O2. We review the conditions, discussed earlier by Cooper, under which the flux of a component in a multicomponent system becomes proportional to its concentration gradient (Fickian diffusion), as if the entire solvent matrix behaves as a single component, and also suggest a method of incorporating the thermodynamic effect on diffusion in the same spirit. Regardless of the magnitude or sign of the off-diagonal terms of the D matrix, it is always possible to define an effective binary diffusion coefficient (EBDC) of a component in a semi-infinite multicomponent diffusion-couple experiment such that it has the property of the Fickian diffusion coefficient, provided that there is no inflection on the diffusion profiles. It is shown that the success of Elphick et al. in fitting the experimental diffusion profiles of all components over a limited concentration range by a single diffusion coefficient is due to fortuitous similarity of the EBDCs of the components (Fe, Mg, Mn and Ca) in their diffusion couple experiments. In common metapelitic garnets showing compositional zoning, the EBDCs of the divalent cations do not differ from each other by more than a factor of 2.5. However, the EBDC of a component changes from core to rim by a factor of 3 to 12, depending on the composition. We suggest a method of volume averaging of the EBDC which should prove useful in approximate calculations of diffusion flux during relaxation of compositional zoning. The EBDC of Mn is found to reduce essentially to DMnMn, the main diagonal term of the D matrix, and consequently can be calculated quite easily. Evaluation of EBDC of Fe, Mg and Mn in garnets from a prograde Barrovian sequence did not reveal any significant dependence on the extent of relaxation of garnet. The diffusion data have been applied to calculate the cooling rate of natural biotite-garnet diffusion couple from eastern Finland and diffusional modification of growth zoning in garnet in early Proterozoic Wopmay orogen, Canada. The results are in good agreement with geochronological and other independent constraints.

A THEORY OF DIFFUSION IN THE L12 STRUCTURE II: TRACER AND CHEMICAL DIFFUSION

A simple and absolute closed capillary method is introduced for measuring tracer diffusion coefficients in liquids with both γ- and β-active tracers. In this method a narrow capillary is partially filled with a labelled solution and an unlabelled solution is used to fill the rest of the capillary. The diffusion coefficient is obtained from the time dependence of the monitored activity when the length of the capillary is known. The method has been tested by remeasuring the tracer diffusion coefficients of22NaCl and Na36Cl for solutions covering a wide range of total NaCl concentration. The average precision of the measurements was about 0.3%. The results obtained for Na36Cl tracer diffusion are in good agreement with the data found in literature. The22NaCl tracer diffusion coefficients that were measured in dilute solutions agree well with those obtained using the continuous open-ended capillary method but differ from results for solutions between 0.1 and 1M obtained with the diaphragm cell method.

Composition dependence of tracer diffusion coefficients in Fe–Ga alloys: A case study by a tracer-diffusion couple method

Prediction of the mutual diffusivity in binary non-ideal liquid mixtures from the tracer diffusion coefficients

Diffusion in ordered alloys and intermetallic compounds