Numerical study of grain boundary diffusion in nanocrystalline ionic materials including blocking space charges

Diffusion in two grain boundary networks (parallel boundaries and square grains) was considered in order to analyze the effects of space charge layers on diffusion profiles. The space charge zones are supposed to be depleted of mobile charge carriers, leading to a blocking of transport of diffusing species. The finite element method was used to numerically calculate the concentration distribution and the corresponding integrated diffusion profiles. The accuracy of determining the grain boundary diffusivity from the simulated profiles, applying conventional equations was estimated. Blocking space charge layers lead to a possible overestimation of the grain boundary diffusivity in the type-B kinetic regime. Moreover, they lead to a significant shifting of kinetic regimes and extremely large diffusion times should be used to reduce errors, when determining the grain boundary diffusivity in the type-A regime.     

Effect of High-Temperature Structure and Diffusion on Grain-Boundary Diffusion Creep in fcc Metals

Molecular dynamics simulations of high-energy twist and tilt bicrystals of fcc palladium reveal a universal, liquid-like, isotropic high-temperature diffusion mechanism, characterized by a rather low self-diffusion activation energy that is independent of the boundary type or misorientation. Medium-energy grain boundaries exhibit the same behavior at the highest temperatures; however, at lower temperatures the diffusion mechanism becomes anisotropic, with a higher, misorientation-dependent activation energy. Our simulations demonstrate that the lower activation energy at elevated temperatures is caused by a structural transition, from a solid boundary structure at low temperatures to a liquid-like structure at high temperatures. We demonstrate that the existence of such a transition has important consequences for diffusion creep in nanocrystalline fcc metals. In particular, our simulations reveal that in the absence of grain growth, nanocrystalline microstructures containing only high-energy grain boundaries exhibit steady-state diffusion creep with a creep rate that agrees quantitatively with that given by the Coble-creep formula. Remarkably, the activation energy for the high-temperature creep rate is the same as that characterizing the universal high-temperature diffusion in high-energy energy bicrystalline grain boundaries.     

Diffusion of nickel in single- and polycrystalline Cr2O3

Chromia protective layers are formed on many industrial alloys to prevent corrosion by oxidation. Their role is to limit the inward diffusion of oxygen and the outward diffusion of cations. A number of chromia-forming alloys contain nickel as a component, such as steels, FeNiCr and NiCr alloys. To ascertain if chromia is a barrier to outward diffusion, nickel diffusion in chromia was studied in both single crystals and polycrystals in the temperature range 900–1100°C at an oxygen pressure of 10^?4 atm (argon + 100 ppm O₂). A nickel film of ～35 nm thick was deposited on the chromia surface and, after diffusing treatment, nickel penetration profiles were established by secondary ion mass spectrometry (SIMS). Two diffusion domains appear in polycrystals, the first domain is assigned to bulk diffusion and the second is due to diffusion along grain boundaries. For the bulk diffusion domain and diffusion in single crystals, using a solution of Fick's second law for diffusion from a thick film, bulk diffusion coefficients were determined at 900 and 1000°C. At the higher temperature, a solution of Fick's second law for diffusion from a thin film could be used. For the second domain in polycrystals, Le Claire's model allowed the grain boundary diffusion parameter (αD _gb δ) to be established. Nickel bulk diffusion does not vary significantly according to the microstructure of chromia. The activation energy of grain boundary diffusion is slightly greater than the activation energy of bulk diffusion, probably on account of segregation phenomena. Nickel diffusion was compared with cationic self-diffusion and with literature data on Fe and Mn heterodiffusion in the bulk and along grain boundaries. All results were analyzed in relation to the oxidation process of stainless steel.     

Grain Boundary Kinetics in Oxide Ceramics with the Cubic Fluorite Crystal Structure and its Derivatives

Kinetics of grain boundaries in oxides with the cubic fluorite structure and its derivatives has been investigated using fine grain ceramics that are fully dense. Grain growth measurements in these materials have provided information on grain boundary diffusivity over a diffusion distance of the order of the initial grain size. With the addition of solute cations, grain boundary mobility can be varied over many orders of magnitude, often with very different activation energies. This is caused by the variation of defect population and the defect-solute association. Definitive evidence for solute drag has also been observed in some cases, but solute drag can not be confirmed as a general mechanism in solid solutions. Lastly, while grain boundary at low temperature may continue to serve as a fast diffusion path, it may not be able to migrate because of additional pinning mechanisms such as those exerted by grain boundary nodal points or lines. This means that sintering without grain growth is possible, opening up an avenue for obtaining ultrafine ceramics by pressureless sintering.     

Grain boundary interdiffusion in the case of concentration-dependent grain boundary diffusion coefficient

The grain boundary diffusion in a binary system which exhibits a grain boundary phase transition is considered in the framework of Fisher's model. The kinetic law of the growth of the grain boundary phase and the distribution of the diffusant near the grain boundary are calculated. The method of determining of the concentration dependence of the grain boundary diffusion coefficient from the experimentally measured penetration profiles of the diffusant along the grain boundaries is suggested. The experimental results on Zn diffusion in Fe(Si) bicrystals, Ni diffusion in Cu bicrystals and grain boundary grooving in Al in the presence of liquid In are discussed in light of the suggested model.     

Surface exchange reactions and fast grain boundary diffusion in polycrystalline materials: Application of a spherical grain model

Laplace transforms of the solution functions to the diffusion equations for surface exchange reactions and fast grain boundary diffusion in polycrystalline materials of finite thickness have been derived by applying a spherical grain model. Diffusion profiles have been calculated for semi-infinite diffusion systems as well as thin films by application of numerical Laplace inversion. The surface exchange reaction at the surface of the sample (e.g. oxide ceramics) in contact with the constant diffusion source (e.g. gas phase) is assumed to be fairly slow such that the diffusion source is not in equilibrium with the surface during the diffusion anneal. Two limiting cases for the surface conditions are taken into account, viz. fast surface diffusion and a uniform ratio of the surface exchange coefficient/diffusion coefficient. The calculated profiles refer to Harrison's type-A diffusion kinetics. Apart from expressions for the effective diffusion coefficient, analogous relations for the effective surface exchange coefficient are proposed. Relaxation curves for the total amount of diffusant exchanged with the diffusion source are discussed in terms of the diameter of the spherical grains (average grain size), surface exchange coefficient, bulk and grain boundary diffusion coefficient, respectively.     

Theory of bending of polycrystalline beams by creep at low stress

Attempts to extend diffusion creep theory from simple grain geometries to more complex polycrystalline structures generally make the assumptions that the vacancy creation (or annihilation) rate is constant over each grain face and that the volume of each grain is conserved. These assumptions do not permit grain rotation, a common feature of polycrystalline creep, nor is diffusion allowed to occur between individual grains. These two aspects are investigated theoretically in this paper, for the specific case of the grain boundary diffusion controlled bending of a polycrystalline beam consisting of a set of orthorhombic grains of dimensions X, Y and Z, with the Z dimension, parallel to the axis of bending, assumed large such that two-dimensional diffusion predominates. The grains are aligned with continuous boundaries across the beam height. For grains highly elongated along the beam length (X???Y), the derived rotation rate is identical to that for a bicrystal having the same height as the beam. For smaller X, diffusion in boundaries along the beam length make increasing contributions and the rotation rate increases. The novel prediction is made that the non-conservation of grain volume is an inevitable consequence of the grain boundary diffusion controlled deformation of this particular polycrystalline configuration.     

Temperature and orientation dependences of the grain-boundary diffusion coefficient of antimony in copper bicrystals

An investigation is made of the diffusion of antimony through the bulk and along grain boundaries in copper bicrystals containing a symmetric 〈100〉 misorientation boundary with misorientation angles from 20 to 37.2°. The bicrystals are grown by the method of horizontal zone recrystallization. The temperature range for these studies is 480–580 °C, where the solubility of Sb in Cu is about 6 atomic % and practically temperature-independent. The concentration profiles are obtained by x-ray spectral microanalysis, and the grain-boundary diffusion parameters are computed by the method of Whipple and Suzuoka. The orientation dependence of the triple product P=sδD _b (where s is the segregation coefficient, δ the width of the grain boundary, and D _b the grain-boundary diffusion coefficient) is nonmonotonic, with a maximum for the special ∑5 misorientation boundary (36.9°). The effective activation energy for grain-boundary diffusion ranges from ∼70 kJ/mol for ∑5 to140 kJ/mol for general boundaries. Fiz. Tverd. Tela (St. Petersburg) 39, 1153–1157 (July 1997)     

Transient-state method for coupled evaluation of Soret and Fick coefficients,and related tortuosity factors,using free and porous packed thermodiffusion cells: Application to CuSO4 aqueous solution ( 0.25M)

The measurement of Soret coefficients in liquids is not easy and usually not very precise because the resulting concentration gradient is small and moreover can be perturbed by undesired convection currents. In order to suppress, or to drastically reduce these convection currents, the use of a porous medium is sometimes suggested. The question arises as to whether the Soret coefficient is the same in free fluid and in porous medium. This is the aim of this paper. To this end, for a given liquid mixture, the time evolution of the vertical concentration gradient is experimentally measured in the same thermodiffusion cell filled first with the free liquid and next with a porous medium followed by saturation by the liquid mixture. Both the isothermal diffusion (Fick) coefficient and the Soret coefficient can be deduced, providing that a correct working equation is used. The proposed equation results from integration of the general mass conservation equation with realistic boundary conditions (zero mass flux at the boundaries) and some simplifying assumptions rendering this equation more tractable than the one proposed some decades ago by Bierlein (J.A. Bierlein, J. Chem. Phys. 23, 10 (1955)). The method is applied here to an electrolytic solution (CuSO4, 0.25 M) at a mean temperature of 37^°C. The Soret coefficients in free and porous medium (zircon microspheres in the range of 250- 315 ^. 10^-6m) may be considered to be equal ( S_T = 13.2±0.5 ^. 10^-3 K^-1) and the tortuosity factors for the packed medium are the same relative to thermodiffusion and Fick coefficients ( = 1.51±0.02).     

Silver grain boundary diffusion in Pd

Two to ten nanometer thick polycrystalline Pd films were prepared on the (1 1 1) surface of Ag single crystal and investigations of the Ag diffusion along Pd grain boundaries were carried out using the Hwang-Balluffi method. The samples were monitored by Auger electron spectroscopy (AES) during isothermal heat treatments in the 438-563 K temperature range. Using plausible simplifying assumptions, the activation energy of the product of the grain boundary (GB) diffusion coefficient and k′ (k′ = c_s/c_gb; c_s and c_gb are the surface and GB concentration, respectively) was calculated (0.99 ± 0.08 eV) from the evaluated saturation coefficients of the surface accumulation. This energy, for weak temperature dependence of k′, is approximately equal to the activation energy of the GB diffusion.     

The nonlinear Fokker-Planck equation with state-dependent diffusion - a nonextensive maximum entropy approach

Nonlinear Fokker-Planck equations (e.g., the diffusion equation for porous medium) are important candidates for describing anomalous diffusion in a variety of systems. In this paper we introduce such nonlinear Fokker-Planck equations with general state-dependent diffusion, thus significantly generalizing the case of constant diffusion which has been discussed previously. An approximate maximum entropy (MaxEnt) approach based on the Tsallis nonextensive entropy is developed for the study of these equations. The MaxEnt solutions are shown to preserve the functional relation between the time derivative of the entropy and the time dependent solution. In some particular important cases of diffusion with power-law multiplicative noise, our MaxEnt scheme provides exact time dependent solutions. We also prove that the stationary solutions of the nonlinear Fokker-Planck equation with diffusion of the (generalized) Stratonovich type exhibit the Tsallis MaxEnt form. Received 26 February 1999     

Studies on interdiffusion in Pd/Mg/Si films: Towards improved cyclic stability in hydrogen storage

The paper reports the diffusion coefficients of grain boundary diffusion and grain boundary assisted lattice diffusion of Pd in Mg in Pd/Mg/Si system, a useful material for hydrogen storage, at 473 K in vacuum. The grain boundary diffusivity is measured by Whipple model and grain boundary assisted lattice diffusivity by plateau rise method using Pd depth profiles constructed by Rutherford backscattering spectrometry. It is established that grain boundary diffusivities are about six orders of magnitude faster than lattice diffusivities. Fine grained microstructure of Pd film, high abundance of defects in Mg film and higher stability associated with Pd-Mg intermetallics are responsible for the diffusion of Pd into grain boundaries and subsequently in the interiors of Mg. Besides the indiffusion of Pd, annealing also brings about an outdiffusion of Mg into Pd film. Examination by nuclear reaction analysis involving ²⁴Mg(p,p′γ)²⁴Mg resonance reaction shows the occurrence of Mg outdiffusion. Minimization of surface energy is presumably the driving force of the process. In addition to Pd/Mg interface, diffusion occurs across Mg/Si (substrate) interface as well on increasing the annealing temperature above 473 K. These studies show that dehydrogenation of films accomplished by vacuum annealing should be limited to temperatures less than 473 K to minimize the loss of surface Pd, the catalyst of the hydrogen absorption-desorption process and Mg, the hydrogen storing element, by way of interfacial reactions.     

Grain boundary diffusion of Cu in TiN film by X-ray photoelectron spectroscopy

In this study, the grain boundary diffusion of Cu through a TiN layer with columnar structure was investigated by X-ray photoelectron spectroscopy (XPS). It was observed that Cu atoms diffuse from the Cu layer to the surface along the grain boundaries in the TiN layer at elevated temperature. In order to estimate the grain boundary diffusion constants, we used the surface accumulation method. The diffusivity of Cu through TiN layer with columnar structure from 400 °C to 650 °C is D_b≈6×10⁻¹¹exp(−0.29/(k_BT )) cm²/s. Received: 18 May 1999 / Accepted: 8 September 1999 / Published online: 23 February 2000     

Investigation of solid-state reaction in Ag/Sn nanostructured thin films at room temperature

Diffusion intermixing processes in nanostructured Ag/Sn thin-film system at room temperature were investigated by means of secondary neutral mass Spectrometry depth profiling technique. As it was confirmed by X-ray diffraction too, the reaction started already in the as-deposited sample. Since the bulk diffusion was frozen at room temperature, the Ag₃Sn phase was formed along the grain boundaries (GBs), gradually consuming the interior of grains, and was grown perpendicular to the GBs. At the same time, formation and growth of a small compact reaction layer near the interface were observed and the shift of the bordering parallel interfaces was controlled by GB diffusion. From the kinetics of the diffusion process in the above two mechanisms, both the interface velocity in the diffusion-induced grain boundary motion regime as well as the coefficient of parabolic growth in the planar growth regime were determined.     

Correlation among grain boundary character,carbide precipitation and deformation in Alloy 690

The correlation among grain boundary character, carbide precipitation and deformation in the grain boundary engineering (GBE) treated Alloy 690 samples with and without pre-deformation aged at 715^oC for 15?h was analysed by scanning electron microscopy and electron backscatter diffraction. The fraction of low Σ coincidence site lattice (CSL) grain boundary was enhanced by GBE treatment. The fraction of Σ3 grain boundary decreased, and most of Σ9 and Σ27 grain boundaries disappeared in the deformed GBE samples. After aging treatment, bigger carbide precipitated at coherent Σ3 grain boundary, however, most of plate-like carbide precipitated at incoherent Σ3 grain boundary disappeared in the pre-deformed GBE samples. The larger carbide precipitated on the random grain boundary in the 5% pre-deformed sample, while smaller carbide can be observed in the 15% pre-deformed sample. During the in situ tensile test of the aged GBE samples, grain boundary carbide migrated with the grain boundary migration. The slip bands go across Σ3 grain boundary directly, but cannot go across other grain boundaries. The high density of carbide plate precipitated near incoherent Σ3 and Σ9 grain boundaries can resist the evolution of slip bands. Compared to the Σ3 and Σ9 grain boundaries, Σ27 and random grain boundaries are more easily to form microcrack during deformation. The initiation of grain boundary microcrack not only related to the character of grain boundary but also related to the character of nearby grain boundaries. The phase interface of carbide and matrix is another region to initiate the microcrack.     

Computer simulation of interdiffusion in polycrystalline thin film couples

Computer simulation of superimposed lattice, grain boundary and surface diffusion, characteristic for polycrystalline thin film diffusion systems, was performed by way of discretisation of the nonlinear diffusion law. In order to give a vivid impression how such a complex process takes place under some typical conditions we have chosen pseudo three-dimensional computer plots of the spacial distribution of the concentration instead of the commonly used iso-concentration diagrams. The following cases are considered:

The Harrison diffusion kinetics regimes in solute grain boundary diffusion

Knowledge of the limits of the principal Harrison kinetics regimes (Types A, B and C) for grain boundary diffusion is very important for the correct analysis of depth profiles in a tracer diffusion experiment. These regimes for self‐diffusion have been extensively studied in the past by making use of the phenomenological lattice Monte Carlo (LMC) method with the result that the limits are now well established. However, the relationship of these self‐diffusion limits to the corresponding ones for solute diffusion in the presence of solute segregation to the grain boundaries remains unclear. In the present study, the influence of solute segregation on the limits was investigated with the LMC method for the well‐known parallel grain boundary slab model by showing the equivalence of two diffusion models. It is shown which diffusion parameters are useful for identifying the limits of the Harrison kinetics regimes for solute grain boundary diffusion. It is also shown how the measured segregation factor from the diffusion experiment in the Harrison Type‐B kinetics regime may differ from the global segregation factor.     

The stability of triple junctions during the grain boundary diffusion process

The grain boundary diffusion in a system with triple junctions is considered in such a geometry, in which the flows of diffusing atoms meet at the triple line. The solutions of the diffusion equation is given in the frameworks of Fisher's model and under the assumption of quasi-stationary distribution of the diffusing atoms along the grain boundaries. The change of the mechanical equilibrium at the triple junction due to the increase of the concentration of solute atoms is considered. It is shown that under some circumstances the triple junction looses its stability with respect to migration in the direction to the diffusion source. The stability diagrams in the segregation-diffusivity parameter space are plotted.     

Short-Circuit Diffusion in Ceramics

This article discusses grain boundary diffusion in ceramics. It gives a brief review of the experimental data available for ionic oxides and the problems of interpretation associated with it. The fundamental differences between grain boundary diffusion in metals and ceramics are noted. Calculations of the segregation of defects and impurities to grain boundaries are discussed together with methods of calculating diffusion coefficients in these boundaries. New results for alumina and chromia are presented. The problem of defining a grain boundary width is discussed with respect to new calculations on nickel oxide.