Grain Boundary Diffusion Model Research Articles

Anisotropic grain boundary diffusion in binary alloys: Phase-field approach

In this study, we generalize the phase-field model of grain boundary diffusion in polycrystalline materials. Solute transport is described by the modified Cahn-Hilliard equation accounting for diffusion anisotropy, different interaction parameters in grain boundary region and grain bulk, as well as concentration-dependant mobility. The numerical results are provided for solute transport in plain parallel grains and pollycrystalline medium with various directions of fastest (slowest) solute diffusion. This model predicts subdiffusive power-law dependence for propagation of the concentration profile with the exponent varying in a wide range (α=0.25−0.37 for B-type, α=0.45 for C-type kinetics of grain boundary diffusion). Within the proposed phase-field approach, we confirm the validity of Levine-MacCallum conclusion on concentration profile dynamics and modify the ratios for estimation of the effective values of grain boundary mobility.

Two-dimensional model of grain boundary diffusion and oxidation

The grinding of the structure of materials is accompanied by a change in the physical and mechanical properties. This occurs largely due to the accumulation of energy and defects in the structure, which activates the diffusion of impurities contained in the material. The increase in the number of grain boundaries and joints can cause the inelastic behavior of the material, its additional chemical activation. For some metals and alloys this leads to strengthening, while for others it leads to rapid degradation of mechanical properties. Grain boundary diffusion in such materials is the main mechanism of transport of alloying components or harmful impurities, so its study is important. This paper presents a two-dimensional model of grain boundary diffusion in a material with an explicit structure assignment. The model takes into account the presence of chemical transformations that can determine the corrosion mechanisms under operating conditions. For simplicity of calculation the material structure is taken symmetrical and contains two phases: grains and a boundary phase. Diffusion and kinetic parameters of the phases may differ. The model is represented in dimensionless form so that the distances between neighboring grains or the widths of the boundary phase are the same and the grain sizes can vary. Depending on the ratio of phase sizes we can speak about micro- and nanocrystalline structure. The problem was solved numerically using implicit difference scheme and coordinate splitting. Diffusion and kinetic parameters, which are close to the parameters of oxygen grain boundary diffusion in titanium and titanium oxidation, respectively, were taken for the calculations. Integral concentrations reflect the dynamics (kinetics) of oxygen and oxides accumulation over the calculation area. Results showing the role of changes in the oxidation kinetics due to changes in the reaction constants in the phases and the phase size ratio are presented. The model can be useful for assessing the degree of influence of grain boundary diffusion on the oxidation process and the accompanying change in properties, as well as for setting up appropriate experiments.

Phase-field model of grain boundary diffusion in nanocrystalline solids: Anisotropic fluctuations, anomalous diffusion, and precipitation

The anisotropic phase-filed model of grain boundary diffusion and precipitation of solute in nanocrystalline solids has been developed. In this model, the Cahn–Hilliard equation is generalized for the anisotropic phase-field diffusion of solute and anisotropic compositional fluctuations. It is found that dynamics of solute concentration profile demonstrates the anomalous diffusion behavior with scaling parameters depending on the mobility ratio and microstructure of a solid solution. It is noteworthy that the increase in source concentration can slow down the concentration front propagation due to uphill diffusion or formation of a new phase. Parameters of grain boundary diffusion control the precipitation dynamics. In particular, a decrease in transverse diffusion coefficient is responsible for longer incubation time, and lower rates of nucleation and nuclei growth in comparison with the case of isotropic solute transport near grain boundaries. Transport properties of boundary and bulk are responsible for the formation of the bimodal size distribution function of second phase particles and specific kinetics of average radius and number density.

Влияние зернограничной диффузии на окисление сплава Ti3Al

The paper presents a model of grain-boundary diffusion, supplemented by the equations of oxide formation kinetics. Conditions are considered isothermal. The alloy material in the model is represented by alternating intermetallic grains with a clear selection of triple junctions between them. Diffusion and kinetic properties of grains and boundaries may differ. The model is used to describe the process of an intermetallic alloy oxidation in bulk and along grain boundaries. The distributions of the concentrations of elements and oxides, and the change in the average integral concentrations of oxides were studied with varying reaction rate constants in the grains and in the vicinity of the boundaries.

Grain-boundary nitrogen diffusion model in multilayer materials

The paper presents the dependencies for the formation of diffusion layers in materials having a laminar structure, when they are saturated with nitrogen. The influence of chemical composition on the thickness of the nitrided layer has been determined. Models for formation of diffusion layers depending on permeability of interlayer boundaries and chemical formula of the composition are proposed.

An expansion of the Fisher model for concentration dependent grain boundary diffusion

In accounting for concentration dependent grain boundary diffusion, an extra term is derived and added to the boundary condition in Fisher’s Model for atomic transport in fast diffusing boundaries under the condition of low diffusant concentration resulting in an expanded model. The validity of this is evaluated using an implicit finite difference model which shows that a minimum of around 0.5–0.6 order of magnitude difference between the minimum and maximum grain boundary diffusion values is necessary for a significant effect to be observed. This was investigated for a grain boundary diffusion coefficient that exponentially increased and decreased with concentration. Whereas effects on measured grain boundary diffusion coefficients are evaluated to be negligible, the concentration dependency of the grain boundary diffusion substantially affects the elemental distribution in the surrounding of the grain boundary. The latter directly influences grain boundary related metallurgical phenomena such as grain growth or phase transformations in nanocrystalline materials.

Grain-boundary diffusion modeling in a microstructural material

The paper presents a two-dimensional model of grain-boundary diffusion with consideration for triple junctions. A diffusant is assumed to come from an amorphous film containing an active component. In heating, the diffusant is transferred from the film into the crystalline material, which is composed of the two phases: grain (volume) phase and grain-boundary phase. The material structure is specified explicitly, which allows an investigation of the influence of the phase size and triple junctions on diffusion dynamics. Boundary conditions suggest a perfect contact between the phases. Consideration is given to a temperature dependence of the diffusion coefficient. A variation in temperature with time is specified explicitly. The problem is solved numerically. We plot concentration-time dependences at points of the computational domain as well as the diffusant concentration distribution over the phases to illustrate the effect of grain size on diffusion. The diffusant is found to behave nonmonotonously, accumulating and consuming, at the triple junction during diffusion. The higher the diffusion coefficient ratio between the phases and the smaller the grain, the faster the concentration variation at the triple junction.

Realization of adhesion enhancement of CuO nanowires growth on copper substrate by laser texturing

CuO nanowires (NWs) are easily grown by thermal oxidation, but the practical applications have been restricted due to the cracking of CuO NWs films or exfoliation from the copper substrate. In this paper, laser texturing combined with thermal oxidation to synthesize CuO NWs demonstrates that the two-step approach is beneficial to the growth and adhesion enhancement of CuO NWs on the copper substrate. Laser texturing at different laser scan times and scan spacing forms micro-patterned grids on the copper sheet, followed by the thermal oxidation growth of CuO NWs on the laser textured copper sheet. The adhesion of CuO NWs on the copper substrate is evaluated by tape test and ultrasound test. The cross-sectional SEM images of the CuO NWs sample and the element concentrations at different layers indicate that the growth mechanism of CuO NWs conforms to the stress-driven grain-boundary diffusion model. The analysis shows that the formed micro-patterned grids by laser texturing provide additional contact area for copper and thermally activated oxygen. In addition, it contributes to thermal stress release at the edges of micro-grids, which is capable to protect the oxide layer from exfoliation. Therefore, the proposed approach has proved to be effective to adhesion enhancement of CuO NWs on the copper substrate, which is of great significance for promoting and extending the potential applications of CuO NWs in various functional surfaces and devices.

A model for time-dependent grain boundary diffusion of ions and electrons through a film or scale, with an application to alumina

A model for ionic and electronic grain boundary transport through thin films, scales or membranes with columnar grain structure is introduced. The grain structure is idealized as a lattice of identical hexagonal cells – a honeycomb pattern. Reactions with the environment constitute the boundary conditions and drive the transport between the surfaces. Time-dependent simulations solving the Poisson equation self-consistently with the Nernst-Planck flux equations for the mobile species are performed. In the resulting Poisson-Nernst-Planck system of equations, the electrostatic potential is obtained from the Poisson equation in its integral form by summation. The model is used to interpret alumina membrane oxygen permeation experiments, in which different oxygen gas pressures are applied at opposite membrane surfaces and the resulting flux of oxygen molecules through the membrane is measured. Simulation results involving four mobile species, charged aluminum and oxygen vacancies, electrons, and holes, provide a complete description of the measurements and insight into the microscopic processes underpinning the oxygen permeation of the membrane. Most notably, the hypothesized transition between p-type and n-type ionic conductivity of the alumina grain boundaries as a function of the applied oxygen gas pressure is observed in the simulations. The range of validity of a simple analytic model for the oxygen permeation rate, similar to the Wagner theory of metal oxidation, is quantified by comparison to the numeric simulations. The three-dimensional model we develop here is readily adaptable to problems such as transport in a solid state electrode, or corrosion scale growth.

Determination of Grain Boundary Diffusion Parameters Based on Specified Model of Grain Boundary Diffusion and Combined Analysis of Radiotracer and Mössbauer Spectroscopy Data

The classical Fisher model of grain-boundary diffusion and the traditional method of determination of grain-boundary diffusion parameters by radiotracer technique combined with the serial-sectioning method are analyzed. The Fisher model specification based on the data of the emission Mössbauer spectroscopy is considered, and the additional information which can be extracted from the Mössbauer studies is discussed. The possibility of determination of grain-boundary diffusion parameters based on the combined analysis of the radiotracer technique and Mössbauer spectroscopy with the application of the specified Fisher model of grain-boundary diffusion is considered. This approach is demonstrated by an example of determination of grain-boundary diffusion of Co in W and Mo.

Understanding the role of oxygen in the segregation of sodium at the surface of molybdenum coated soda‐lime glass

Molybdenum (Mo) coated soda‐lime glass is a commonly used substrate for Cu(InGa)Se2 solar cells as it also acts as the sodium (Na) source, which improves the efficiency of these devices. In this work, we investigate how oxygen controls the segregation and accumulation of Na on the Mo surface. A direct relationship between the concentration of surface oxygen and the amount of Na accumulation is showed. Values for the surface segregation ratio and grain boundary diffusion coefficient for Na in Mo are obtained by fitting diffusion data at several temperatures to a model for grain boundary diffusion. The results of this model reveal that surface oxygen controls the Na saturation level through its effect on the surface segregation of Na. An activation energy for grain boundary diffusion of Na is estimated and is similar to that of MoO bond dissociation in MoO3 suggesting the involvement of this bond during Na transport. © 2014 American Institute of Chemical Engineers AIChE J, 60: 2365–2372, 2014

Effect of atomic interaction on grain boundary diffusion in the B regime

Atomic interaction in grain boundaries that leads to complex formation in two-component solid solutions A–B and the effect of such complexes on grain boundary diffusion are discussed. Two particular types of complex are considered: AB, for systems with restricted solubility and intermediate phases of the AB type, and B2, for systems with restricted solubility without intermediate phases. A thermodynamic description of the complex formation is discussed in the framework of a model of ideal associated solution. Assuming that the complexes are immobile in grain boundaries but that atoms within the complexes are involved in the leakage to the bulk, a new numerical solution of Fisher’s model of grain boundary diffusion is obtained. The solution shows that the complex formation leads to a decrease in diffusion flux along a grain boundary. The theory is in good agreement with several experimental results found in the literature.

Determination of Grain Boundary Diffusion Parameters Based on Combined Analysis of Radiotracer Technique and Mössbauer Spectroscopy Data

A method of grain boundary diffusion characterization has been suggested based on combined analysis of radiotracer serial-sectioning technique and Mössbauer spectroscopy data and a specified model of grain boundary diffusion. Its application for determination of parameters of Co grain boundary diffusion in W is demonstrated.

Determination of the parameters of grain-boundary diffusion and segregation of Co in W using an improved model of grain-boundary diffusion

Based on the recently proposed model of grain-boundary diffusion, the available research data on grain-boundary diffusion of Co in W, and on emission Mossbauer studies of grain boundaries in polycrystalline W have been analyzed in detail. Using joint processing of primary data for tracer analysis and Mossbauer studies, all the parameters of grain-boundary diffusion of Co in W have been determined: the coefficient of grain-boundary segregation, coefficient of grain-boundary diffusion, and the diffusion width of the grain boundary.

The potential to use fission gas release experiments to measure lattice and grain boundary diffusion in metallic fuels

We have applied a model for lattice and grain boundary diffusion in polycrystalline materials to assess the potential for the use of fission gas release experiments to measure the lattice and grain boundary diffusion coefficients in metallic nuclear fuel materials. Our assessment is that, assuming that grain boundary diffusion in metallic fuels is similar to that in other metals, it is reasonable to expect that lattice diffusion coefficients can be determined from short time gas release experiments and the product of the grain boundary diffusion coefficient, the segregation factor, and the boundary width can be extracted from gas release experiments at longer times. Under the same assumption, activation energies can be deduced from the temperature dependence of the measured diffusivities.

Diffusion at Short Circuits: State of the Art

Evidence for solid-state diffusion (the second half of the 19th century). The first measurements of solid state diffusion (W. Roberts-Austen, 1896–1922). The first tracer experiments to determine the solid-state diffusion (G. von Hevesy, 1913–1923). The first evidence of accelerated diffusion in polycrystalline materials (1924–1935). Autoradiographic studies of grain boundary diffusion (50s of 20th century). The first quantitative experimental and theoretical studies of the “short circuiting” diffusion (beginning from 1949, D. Turnbull and R. Hoffman – General Electric Research Lab.): radiotracer serial sectioning method, the Fisher model (1951) for grain boundary diffusion, exact solutions and developments of the Fisher model (1954–1963). The progress in the experimental methods for determination of grain boundary diffusion data and results of measurements for different metallic systems (up to date). The measurements of grain boundary diffusion parameters in the B and C regimes. Grain boundary diffusion and grain boundary segregation. Nonlinear segregation effects. Structural effects of grain boundary diffusion. Diffusion in bicrystals. Diffusion in nanocrystals. Computer simulation of grain boundary diffusion. Mechanisms of grain boundary diffusion.

Grain boundary diffusion patterns under nonequilibrium and migration of grain boundaries in nanoctructure materials

In this work, the model of grain boundary diffusion from a permanent source along nonequilibrium migratory grain boundaries is considered. Grain boundary nonequilibrium is characterized by a value of boundary excess energy up to which relaxation goes. It is shown increasing excess energy and migration velocity of nonequilibrium boundaries lead to increasing diffusant volume penetrating into a sample during annealing time.

A model of grain boundary diffusion in polycrystals with evolving microstructure

Abstract We consider grain boundary diffusion in a polycrystal in which normal grain growth occurs simultaneously with diffusion. We assume that any point within the diffusion zone is swept a number of times by migrating grain boundaries, and develop a model which operates with the average concentrations of the diffusant in the planes parallel to diffusion front. We derive analytical expressions describing the diffusion profiles both for constant source and instantaneous source boundary conditions. Finally, we suggest a practical method for extracting the grain boundary diffusion coefficient from experimentally measured diffusion profiles. This method relies on microstructure characterization and requires the grain growth constant and grain growth exponent to be determined in the independent set of experiments.

Growth strains and creep in thermally grown alumina: Oxide growth mechanisms

In situ measurements of growth strains and creep relaxation in α-Al2O3 films, isothermally grown on β-NiAl alloys at 1100 °C, are reported and analyzed. Samples containing the reactive element Zr, and Zr-free samples, are examined. For Zr-free samples, steady state growth strains are compressive, whereas the growth strains are tensile when the reactive element (RE) is added to the alloy. This behavior is attributed to the counterflow of oxygen and aluminum interstitials, and to simultaneous counterflow of oxygen and aluminum vacancies, all moving through the grain boundaries. Cross diffusing oxygen and aluminum interstitials may merge and combine within the film, forming new oxide along grain boundary walls, a mechanism that leads to an in-plane compressive stress. Cross diffusing oxygen and aluminum vacancies will also merge and combine within the film; in this case material is removed from grain boundary walls, a mechanism that leads to an in-plane tensile stress. When no RE is present, the interstitial mechanism dominates and the resultant stress is compressive. Consistent with the “dynamic segregation model,” the RE slows the outdiffusion of Al interstitials permitting the tensile mechanism to dominate. This interpretation invokes the unconventional view that oxygen and aluminum interstitials and vacancies, created in and driven by the strong chemical gradient, all participate meaningfully in the scale growth process. Grain boundary diffusion measurements were obtained from low stress creep data, interpreted using the Coble model of grain boundary diffusion. Reported diffusion measurements of oxygen through grain boundaries of α-Al2O3, which are known to be inconsistent with oxide scale growth, are critically examined. A simple picture, a “balanced defect model,” emerges that is consistent with the dynamic segregation model, observed growth stresses and their dependence on the presence of a reactive element, sequential oxidation experiments, and our best knowledge about grain boundary diffusion coefficients.

A Mathematical Formulation for Interfacial Diffusion, Incorporating Deviation from the Classical Random Walk Theory

Fisher’s model for grain boundary diffusion considers the lattice and the grain boundary on the same basis by presuming the validity of Fick’s second law for both cases, despite the significant structural differences between them. Recent studies [1-3] have, however, shown that grain boundary diffusion is profoundly different from lattice diffusion. We propose an alternative mathematical formulation that incorporates these structural differences and consequently models grain boundary diffusion phenomena more accurately than Fisher’s model. This is achieved by considering possible deviations from the classical random walk for solute atoms diffusing through grain boundaries. This formalism can also be applied to surface diffusion and triple junction diffusion.