Integrated Diffusion Coefficients Research Articles

Quantifying the diffusion of suprathermal electrons by whistler waves between 0.2 and 1 AU with Solar Orbiter and Parker Solar Probe

Context. The evolution of the solar wind electron distribution function with heliocentric distance exhibits different features that are still unexplained, in particular, the fast decrease in the electron heat flux and the increase in the Strahl pitch angle width. Wave-particle interactions between electrons and whistler waves are often proposed to explain these phenomena. Aims. We aim to quantify the effect of whistler waves on suprathermal electrons as a function of heliocentric distance. Methods. We first performed a statistical analysis of whistler waves (occurrence and properties) observed by Solar Orbiter and Parker Solar Probe between 0.2 and 1 AU. The wave characteristics were then used to compute the diffusion coefficients for solar wind suprathermal electrons in the framework of quasi-linear theory. These coefficients were integrated to deduce the overall effect of whistler waves on electrons along their propagation. Results. About 110 000 whistler wave packets were detected and characterized in the plasma frame, including their direction of propagation with respect to the background magnetic field and their radial direction of propagation. Most waves are aligned with the magnetic field and only ∼0.5% of them have a propagation angle greater than 45°. Beyond 0.3 AU, it is almost exclusively quasi-parallel waves propagating anti-sunward (some of them are found sunward but are within switchbacks with a change of sign of the radial component of the background magnetic) that are observed. Thus, these waves are found to be Strahl-aligned and not counter-streaming. At 0.2 AU, we find both Strahl-aligned and counter-streaming quasi-parallel whistler waves. Conclusions. Beyond 0.3 AU, the integrated diffusion coefficients show that the observed waves are sufficient to explain the measured Strahl pitch angle evolution and effective in isotropizing the halo. Strahl diffusion is mainly attributed to whistler waves with a propagation angle of θ ∈ [15.45]°, although their origin has not yet been fully determined. Near 0.2 AU, counter-streaming whistler waves are able to diffuse the Strahl electrons more efficiently than the Strahl-aligned waves by two orders of magnitude.

Diffusion behaviors and mechanical properties of binary Mg–Ce system

In this work, the solid-solid diffusion reaction and growth kinetics of the Mg–Ce system was investigated by diffusion coupling technique. Four intermetallic compounds Mg12Ce, Mg41Ce5, Mg3Ce, MgCe and one solid solution phase δCe were observed in the diffusion zone with the annealing temperature range of 673–823 K. The results shown that the growth process in the diffusion area was dominated by bulk diffusion. The growth constants, integrated diffusion coefficients, growth activation energies and diffusion activation energies of the four intermetallics were calculated based on the thickness of the phases, reaction temperature and annealing time. Mg3Ce has the largest diffusion activation energy. The mechanical properties of the four IMCs were investigated by using nanoindentation characterization technology and first-principles calculations. The nanoindentation tests show that the Mg12Ce phase possessed the largest Young's modulus (66.08 GPa) and hardness (2.14 GPa). The experimental results are consistent with the trend of the calculated results. The obtained diffusion kinetical properties and mechanical properties of the Mg–Ce system provide useful data for the development of new magnesium-rare earth alloys.

Understanding Interfacial Reactions in Ti-Ni Diffusion Couple.

The diffusion phenomenon in the Ti-Ni binary system was investigated at a temperature of 1173 K. Microstructure and texture analysis revealed the formation of three stable intermetallic compounds, namely Ti2Ni, TiNi, and TiNi3, as well as two metastable intermetallic compounds, including Ti3Ni4 and Ti2Ni3, at the interfacial diffusion zone. The nucleation surface energy increase was analytically estimated, and marker experiments were conducted using thoria particles, both of which showed that Ti2Ni was the first compound to form at the Ti-Ni diffusion interface. At a temperature of 1173 K, using the Wagner method, the integrated diffusion coefficients for the Ti2Ni, TiNi, and TiNi3 phases were calculated to be 3.53 × 10-12, 18.1 × 10-15, and 6.2 × 10-15 m2/s, for, respectively.

Quantitative analysis of diffusion kinetics of intermetallic formation in Ni–Ti system

The diffusion couple method is an appreciated experimental approach in taking in phase relations in multi-component systems. To investigate the growth mechanism of the phases and the diffusion of the components, Ni–Ti diffusion couple experiments are carried out at four different temperatures of 750, 800, 850, and 900 °C up to 24 h. Microstructural observations using scanning electron microscopy equipped with energy dispersive spectroscopy at the interface of the nickel-titanium indicates formation of three major intermetallic compounds by the sequence of NiTi2, Ni3Ti and NiTi. Using typical marker tests with incremental diffusion couples, data about mobilities of elemental type intermetallic phase media can be attained, and tracer diffusion coefficients can be presumed. By using of cerium oxide particles as an inert marker, it is shown that the diffusion of nickel atoms toward titanium is faster than the diffusion of titanium atoms through the compound layers. The marker plane is detected at the interface of Ni/Ni3Ti.The ratio of intrinsic diffusivities indicates that Ni is the predominant diffusing element in this system. The activation energies for the formation and growth of NiTi2, NiTi, Ni3Ti, and the total interfacial layers are found to be respectively 204, 151, 38, and 154 kJ/mol. In addition, the integrated diffusion coefficients are calculated by use of Wagner method. The activation energies for the integrated diffusion coefficient of NiTi2, NiTi and Ni3Ti are calculated as 168,149 and 78 kJ/mol, respectively.

Diffusional behaviors and mechanical properties of Ni-Zn system

In this paper, the solid-solid diffusion reaction and growth kinetics of Ni-Zn system were investigated using diffusion couple technique. Three intermetallic compounds (IMCs), β1 (NiZn), γ (Ni2Zn11) and δ (Ni3Zn22) were observed at the diffusion zone with the annealing temperature range between 598 K and 673 K. The results indicate that the growth process of reaction zone was dominated by volume diffusion. And the growth constants, integrated diffusion coefficients, growth activation energies and diffusion activation energies of γ and δ were evaluated based on thickness of the phases, reaction temperatures and annealing times. The Young’s modulus and Vickers hardness of the three phases were evaluated from the displacement-loading curve by nanoindenter testing. The Young's modulus and hardness of the IMCs increased firstly and then decreased with the increasing of Zn concentration, where γ phase has the highest Young's modulus (183.3 GPa) and highest hardness (6.05 GPa) among the three phases. Finally, the bulk modulus (B), shear modulus (G), Young's modulus (E), and Poisson's ratio (v) of the Ni-Zn IMCs were calculated by the first-principles calculations. The tendency of calculated Young's modulus varying with the composition of Zn was consistent with that of experimental results.

Anomalous growth of Al8Mo3 phase during interdiffusion and reaction between Al and Mo

Molybdenum (Mo) has been reported as a potential candidate for the diffusion barrier between U-10Mo fuel and AA6061 cladding in monolithic fuel plates. Interdiffusion and reaction between Al and Mo was investigated using solid-to-solid diffusion couples annealed in the temperature range from 425° to 625 °C. Thermo-kinetic parameters, i.e. growth rate and integrated diffusion coefficients, were determined using Wagner’s approach. At low temperatures (T ≤ 450 °C), negligible diffusion interaction was observed between Al and Mo. At intermediate temperatures (T = 475° - 600 °C), diffusion couples were observed to mainly develop a layer of Al8Mo3 intermetallic phase, along with a thin, discontinuous layer of Al22Mo5 intermetallic phase between the terminal Al and Al8Mo3 layer. At high temperature (T = 625 °C), continuous layers of both of Al8Mo3 and Al22Mo5 intermetallic phases developed in the diffusion couple. The Al8Mo3 phase grew with a needle-like, non-planar morphology at intermediate temperatures, contrary to the planar growth of the intermediate phase in binary diffusion couple as per Gibbs phase rule. In general, the degree of non-planarity of needle-like Al8Mo3 phase increased with an increase in annealing time at 500 °C, but decreased with an increase in annealing temperature. This non-planar growth was simulated based on three-dimensional anisotropic diffusion using a quadric representation of the diffusivity tensor.

Formation of Magnesium Silicide in bulk diffusion couples

Formation of Magnesium Silicide in binary Mg/Si diffusion couples and in reactions of Mg with Silicon Dioxide was investigated. The position of the Kirkendall marker plane inside the reaction zone of the bulk diffusion couples revealed that at 500 °C, Mg is virtually the only mobile species in the Mg2Si intermetallic compound. Such diffusion behaviour was rationalized using crystallographic information about anti-fluorite structure of the Mg2Si phase. Using general relations between integrated diffusion coefficient for a stoichiometric compound, tracer diffusivities of the components and data on thermodynamic stability of the phases involved in the interaction, the tracer diffusion coefficient DMg⁎ in the Mg2Si was derived.During diffusion-controlled interaction of Mg with Silicon Oxide, the Mg2Si intermetallic is formed within the transition zone microstructure as a continuous phase with the periodically arranged bands of MgO-particles embedded in the matrix phase. In the case of bulk Mg/quartz-SiO2 couples annealed at 500 °C, MgSiO3 and Mg2SiO4 silicates are also found in the reaction zone next to the SiO2-substrate. This can be explained with the help of the equilibrium Mg–Si–O phase diagram. However, no formation of Magnesium Silicates was observed in diffusion-controlled reactions between Mg and fused (amorphous) Silica.At the early stages of the interaction in bulk Mg/SiO2 couples, the situation was found to be different. No quasi-binary compounds MgSiO3 and Mg2SiO4 were detected in the diffusion zone after reaction of Mg with monocrystalline SiO2 as well as fused Silica at 550 °C. In both cases, the layer adjoining the SiO2-substrate is Magnesium Oxide.The formation of periodic layered morphology is considered to be a manifestation of the Kirkendall effect accompanying reactive phase formation in the solid state. The suggested mechanism is operative when the components have distinctly different mobilities in adjacent reaction product layers.

Diffusion Rates of Components in Metal-Silicides Depending on Atomic Number of Refractory Metal Component

Interdiffusion studies conducted in group IVB, VB and VIB metal-silicon systems are discussed in detail to show a pattern in the change of diffusion coefficients with the change in atomic number of the refractory metal (M) component. MSi2and M5Si3phases are considered for these discussions. It is shown that integrated diffusion coefficients increase with the increase in atomic number of the refractory component when the data are plotted with respect to the melting point normalized annealing temperature. This indicates the increase in overall defect concentration facilitating the diffusion of components. This is found to be true in both the phases. Additionally, the estimated ratios of tracer diffusion coefficients indicate the change in concentration of antisite defects in certain manner with the change in atomic number of the refractory components.

Development of different methods and their efficiencies for the estimation of diffusion coefficients following the diffusion couple technique

The interdiffusion coefficients are estimated either following the Wagner's method expressed with respect to the composition (mol or atomic fraction) normalized variable after considering the molar volume variation or the den Broeder's method expressed with respect to the concentration (composition divided by the molar volume) normalized variable. On the other hand, the relations for estimation of the intrinsic diffusion coefficients of components as established by van Loo and integrated diffusion coefficients in a phase with narrow homogeneity range as established by Wagner are currently available with respect to the composition normalized variable only. In this study, we have first derived the relation proposed by den Broeder following the line of treatment proposed by Wagner. Further, the relations for estimation of the intrinsic diffusion coefficients of the components and integrated interdiffusion coefficient are established with respect to the concentration normalized variable, which were not available earlier. The veracity of these methods is examined based on the estimation of data in Ni–Pd, Ni–Al and Cu–Sn systems. Our analysis indicates that both the approaches are logically correct and there is small difference in the estimated data in these systems although a higher difference could be found in other systems. The integrated interdiffusion coefficients with respect to the concentration (or concentration normalized variable) can only be estimated considering the ideal molar volume variation. This might be drawback in certain practical systems.

Interface Reactions and Diffusion of Si3N4 / Ti and Si3N4 / TiAl Alloys

Si3N4 is a ceramic material attracting attention in many fields because of its excellent abrasion resistance. In addition, Ti and TiAl alloys are metals used in a variety of high temperature environments, and have attracted much attention because of their high strength and high melting points. Therefore, study of the interface reaction between Si3N4 / Ti and Si3N4 / TiAl can be a useful practice to identify phase selection and diffusion control. In this study, Si3N4 / Ti5Si3 + TiN / TiN / Ti diffusing pairs were formed in the Si3N4 / Ti interfacial reaction and Si3N4 / TiN(Al) / Ti3Al / TiAl diffusion pathway was identified in the Si3N4 / TiAl interfacial reaction. The diffusion layers of the interface reactions were identified and, to investigate the kinetics of the diffusion layer, the integrated diffusion coefficients were estimated.

Microstructural development of prior nickel coated hot dipped galvanised coatings

Hot dip galvanising of steel is an age old process to protect the steel substrate from corrosion owing to the sacrificial protection from zinc. The galvanised coating quality and performance can be further improved by addition of other alloying elements in molten metal bath or by using prior coatings on the steel substrate. In the present study pure nickel was electrodeposited on Interstitial Free (IF) steel substrate, which was then galvanised. The microstructural evolution of different phases was studied as a function of prior nickel coating thickness and time of galvanising. Glow Discharge Optical Emission Spectroscopy (GDOES) was employed to find out the elemental concentration depth profile and the overall thickness of galvanised coatings. Cross sectional Scanning Electron Microscope (SEM) and Energy Dispersive Spectroscopy (EDS) results revealed the morphology of different phase layers. Additionally, cross sectional elemental mappings were generated using Electron Probe Micro Analyzer (EPMA) and Wavelength Dispersive Spectroscopy (WDS). Different coating phases were identified by grazing angle X-Ray Diffraction (XRD) techniques. The hardness and modulus of individual Ni-Zn phases were obtained from nano indentation method. It was found that application of a prior nickel coating drastically reduced the coating thickness as compared to conventional galvanised coatings. Moreover beyond a certain thickness of prior nickel coating, Ni-Zn phases rather than the Fe-Zn phases, were formed. The dissolution behaviour of nickel in molten zinc bath and subsequent Ni-Zn phase transformation was computed considering the Wagner's approach of integrated diffusion coefficient. For lower time of galvanising the calculated phase layer thicknesses were in well accordance with the experimental data. The hardness of overlay zinc was the least among different phase layers with an increasing trend in nickel rich phases towards the interface.

On the interdiffusion in multilayered silicide coatings for the vanadium-based alloy V-4Cr-4Ti

To provide protection against corrosion at high temperatures, silicide diffusion coatings were developed for the V-4Cr-4Ti alloy, which can be used as the fuel cladding in next-generation sodium-cooled fast breeder reactors. The multilayered coatings were prepared by halide-activated pack cementation using MgF2 as the transport agent and pure silicon (high activity) as the master alloy. Coated pure vanadium and coated V-4Cr-4Ti alloy were studied and compared as substrates. In both cases, the growth of the silicide layers (V3Si, V5Si3, V6Si5 and VSi2) was controlled exclusively by solid-state diffusion, and the growth kinetics followed a parabolic law. Wagner's analysis was adopted to calculate the integrated diffusion coefficients for all silicides. The estimated values of the integrated diffusion coefficients range from approximately 10−9 to 10−13 cm2 s−1. Then, a diffusion-based numerical approach was used to evaluate the growth and consumption of the layers when the coated substrates were exposed at critical temperatures. The estimated lifetimes of the upper VSi2 layer were 400 h and 280 h for pure vanadium and the V-4Cr-4Ti alloy, respectively. The result from the numeric simulation was in good agreement with the layer thicknesses measured after aging the coated samples at 1150 °C under vacuum.

A combined diffusion and thermal modeling approach to determine peak temperatures of thermal metamorphism experienced by meteorites

Carbonaceous chondrites are affected to different degrees by thermal and aqueous metamorphism on their parent bodies. However, the degree of alteration has been categorized mainly by relative scales and achieving quantitative information about metamorphic temperature by conventional mineral thermometry is problematic for low petrologic types. We have developed a general approach to estimate the metamorphic peak temperature experienced by type 3 chondrites from diffusion zoning in minerals, and have applied this approach to olivine in type I and type II chondrules of CO3 chondrites.To obtain metamorphic temperatures from diffusion zoning, we have combined diffusion modeling with thermal modeling of the meteorite parent body. The integrated diffusion coefficient over time (Γ) was identified as a useful parameter to quantify the extent of chemical change by diffusion occurring in a mineral during a given thermal history. Knowing the temperature dependence of the diffusion coefficient, Γ values can be calculated for each thermal history and be compared to the Γ values obtained from diffusion modeling. For thermal histories realistic for the parent body, Γ depends primarily on the metamorphic peak temperature, so that Γ values determined from diffusion profiles in meteorite minerals can be directly related to the metamorphic peak temperature. This general approach is relatively insensitive to uncertainties in the input parameters for the thermal model.We found that chemical zoning in type I and type II chondrule olivine of the CO chondrites Kainsaz and Lancé was largely influenced by solid state diffusion, which is evident from the observed correlation of zoning anisotropy with the crystallographic orientation. Chemical zoning in type II chondrule olivine is mainly igneous for CO chondrites of petrologic types up to at least 3.2 (Kainsaz) and was influenced only minor by diffusion during parent body metamorphism. Fe–Mg zoning in type II chondrule olivine and around sealed cracks in type I chondrule olivine yields similar Γ values, indicating a formation of both zoning features during a common thermal history on the parent body. In addition, Γ values for type II chondrule olivine correlate with metamorphic grade. The application of this approach on Fe–Mg zoning in type II chondrule olivine of CO3 chondrites yields estimates of maximum metamorphic peak temperatures ranging from 653 to 849K for different petrologic subtypes. The Fe–Mg zoning of type I chondrule olivine is not consistent with the peak temperature estimates from type II chondrule olivine, suggesting an additional contribution of solar nebular processes to type I chondrule olivine zoning prior to accretion into the parent body.

Investigation of diffusion behavior in Cu–Sn solid state diffusion couples

The diffusion behaviors and diffusion parameters of intermetallic compounds (IMCs) formed in Cu–Sn diffusion couples were investigated at the temperature range of 130 °C–200 °C. Interdiffusion coefficients of IMCs were calculated based on the measured composition profiles of the diffusion zones. Considering the wavy type of the diffusion layers and the narrow homogeneity range of the IMCs (Cu3Sn and Cu6Sn5), the integrated method was performed to evaluate the integrated interdiffusion coefficients based on the measured thicknesses of the IMCs layers. The transient initial growth stage was excluded by considering two diffusion times where the growths of both IMCs are in the diffusion control stage. The activation energies for diffusion of the IMCs were evaluated from the integrated diffusion coefficients. The growth behavior of Cu3Sn suggested the existence of a transient growth regime for Cu3Sn at the initial stage in cold-bonded Cu–Sn diffusion couple. The intrinsic diffusion coefficients of Cu and Sn in Cu6Sn5 were estimated based on the integrated diffusion coefficients. Sn was found to be the faster diffusion component in the Cu6Sn5 phase. Phase-field simulations combined with the experimentally measured diffusion coefficients and steady-state growth-rate coefficients were performed to estimate the homogeneity range of IMCs. The estimated results are consistent with the experimental results from this study as well as those from literature experimental values and showed that the homogeneity ranges of Cu3Sn and Cu6Sn5 phases in the Cu–Sn system are almost temperature independent.

Interdiffusion in the Ni-Zr System

Interdiffusion study is conducted in the Ni-Zr system in the temperature range of 750-850 A degrees C. Five intermetallic compounds, Ni5Zr, Ni7Zr2, Ni10Zr7, NiZr and NiZr2 are found to grow in the interdiffusion zone. Integrated diffusion coefficients are estimated for different phases. Activation energy is found to be lowest for Ni10Zr7 (178 +/- A 8 kJ/mol) and highest for NiZr (323 +/- A 6 kJ/mol). Tracer diffusion coefficients are estimated at the location of the Kirkendall marker plane in the Ni7Zr2 phase. Ni is found to have 10 times higher diffusion rate compared to Zr in this phase. Higher diffusion rate of Ni compared to Zr is found, which is expected because of higher number of Ni-Ni bonds compared to Zr-Zr bonds in this phase.

Reactive Diffusion in the Re-Si System

A study on reactive diffusion is conducted in the Re-Si system. According to the study, ReSi1.8 phase grows with much higher thickness than the Re2Si phase, in the interdiffusion zone of bulk diffusion couples. The activation energy for integrated diffusion of ReSi1.8 is estimated to be 605 +/- 23 kJ/mol. The growth of the Re2Si phase is studied by considering an incremental diffusion couple of Re/ReSi1.8. Analysis based on the calculation of integrated diffusion coefficients indicates the reason underlying the observed high difference between the growth rates of the ReSi1.8 and Re2Si phases.

An Overview of the Interdiffusion Studies in Mo-Si and W-Si Systems

The growth of phases by reactive diffusion in Mo-Si and W-Si systems are compared. The crystal structures of MSi2 and M5Si3 phases (M = Mo, W) are similar in these two systems. However, the diffusion rates of the components change systematically with a change in the atomic number. Integrated diffusion coefficients in both phases increase with an increasing atomic number of refractory elements i.e. from Mo to W. On the other hand, the ratio of diffusivities of the components decreases. This indicates a relative increase in the diffusion rates of the metal components with increasing atomic number and a difference in defects concentrations in these two systems.

Growth of hafnium and zirconium silicides by reactive diffusion

Diffusion controlled growth of the phases in Hf–Si and Zr–Si systems are studied by bulk diffusion couple technique. Only two phases grow in the interdiffusion zone, although several phases are present in both the systems. The location of the Kirkendall marker plane, detected based on the grain morphology, indicates that disilicides grow by the diffusion of Si. Diffusion of the metal species in these phases is negligible. This indicates that vacancies are present mainly on the Si sublattice. The activation energies for integrated diffusion coefficients in the HfSi2 and ZrSi2 are estimated as 394 ± 37 and 346 ± 34 kJ mol−1, respectively. The same is calculated for the HfSi phase as 485 ± 42 kJ mol−1. The activation energies for Si tracer diffusion in the HfSi2 and ZrSi2 phases are estimated as 430 ± 36 and 348 ± 34 kJ mol−1, respectively.

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.

Diffusion in tungsten silicides

Experiments are conducted in the W-Si system to understand the diffusion mechanism of the species. The activation energies from integrated diffusion coefficients are calculated as 152 ± 7 and 301 ± 40 kJ/mol in the WSi2 and W5Si3 phases, respectively. In both the phases, Si has a much higher diffusion rate compared to W. This is not surprising to find in the WSi2 phase, if we consider the number of nearest neighbors for both the elements in the crystal. The diffusion of W in this phase indicates the presence of W antisites. The faster diffusion rate of Si in the W5Si3 phase indicates the presence of higher concentration of vacancies on the Si sublattice compared to W sublattice.