Grain Boundary Diffusion and Segregation in Interstitial Solid Solutions Based on BCC Transition Metals: Carbon in Niobium

Bulk and grain boundary (GB) diffusion of ¹⁴C in Nb has been studied by the radiotracer serial sectioning technique. B and C kinetic regimes were realized for GB diffusion in the temperature range from 800 to 1173 K. The values of P = sD _gb, D _gb and s follow the Arrhenius dependencies: P = 5.15 × 10^–15 exp[–(83.1 kJ/mol)/RT] m³/s (973–1173 K), D _gb = 2.3 × 10^–6 exp[–(133.0 kJ/mol)/RT] m²/s (800–950 K), and s = 4.7 exp[(49.9 kJ/mol)/RT].The increase in the GB diffusion compared with self-diffusion is very large despite the probable retardation effect due to the strong segregation.The results for GB diffusion of C in Nb as well as for other interstitial solutes (P, S) in bcc transition metals (- Fe, Mo) are discussed in the framework of the transition state theory. It is assumed that GB segregation decreases the energy of the ground state whereas the change in the diffusion mechanism (e.g. from vacancy to interstitial) leads to a strong decrease of the transition state energy. This change in the diffusion mechanism results in a fast GB diffusion of interstitial solutes in spite of their large tendency to segregate to GBs.     

Grain Boundary Motion during Ag and Cu Grain Boundary Diffusion in Cu Polycrystals

Grain boundary (GB) motion in high-purity Cu material (5N8 and 5N Cu) is investigated using the results of radiotracer GB diffusion measurements with tracers exhibiting fundamental differences in the solute-matrix atom interactions. The results on GB solute diffusion of Ag (revealing a miscibility gap in the Ag-Cu phase diagram) and Au (forming intermetallic compounds with Cu) in Cu and on Cu self-diffusion are analyzed.The initial parts of the Ag and Cu penetration profiles turned out to be substantially curved. The profile curvature is explained via the effect of GB motion during ^110m Ag and ⁶⁴Cu GB penetration. The activation enthalpies of GB motion in these two independent measurements occurred to be very close, 95 and 103 kJ/mol, respectively. Moreover, these values turn out to be close, but still somewhat larger than the activation enthalpy of Cu GB self-diffusion in Cu material of the same very high purity, Q ^Cu _gb = 72 kJ/mol. Although tracer diffusion measurements of Au GB diffusion in Cu yielded only limited information on GB motion, the absolute values of GB velocities are consistent with those calculated from the Ag and Cu GB diffusion data.     

The Effect of Pressure on Indium Diffusion along 〈001〉 Tilt Grain Boundaries in Copper Bicrystals

The effect of pressure on the diffusion of indium along two grain boundaries (GBs) in copper bicrystals has been investigated. The GBs studied were of symmetrical tilt type with a 001 misorientation axis. The tilt angle of one GB was 36.5°, which corresponds to the near 5 coincidence orientation, where is the inverse density of the coincidence sites in the two misoriented crystal lattices. The other GB was a general boundary with a tilt angle of 45°. The diffusion along the 001 tilt axis has taken place at the temperature of 923 K and under argon gas pressures up to 1 GPa. The following activation volumes have been found: 0.94±0.11 _Cu for bulk diffusion,–0.5 ±0.7 _Cu for the diffusivity of the near 5 GB and –0.6±0.3 _Cu for the diffusivity of the 45° 100 GB, where _Cu is the atomic volume of copper. Two interpretations of this result seem plausible. The first interpretation is that the In atoms diffuse along the 001 tilt axis by an instertitialcy mechanism. It is known that for such a diffusion mechanism the activation volume is close to zero. The alternative explanation bases on the observation that the parameter measured is the GB diffusivity, which depends on the product of the GB diffusion coefficient and the segregation factor. An enhancement of the In segregation under pressure would lead to an increase of the GB diffusivity. It seems plausible that both interpretations are true and the fact that the atomic volume of In is a factor of 2.2 larger than that of Cu plays an important role both for the increased segregation of In under pressure and the diffusion mechanism.     

Grain Boundary Channel Formation During Interdiffusion in Alloys

We present results of an experimental study of a new phenomenon accompanying grain boundary (GB) interdiffusion: the hole channel formation along GBs. The objects for study were plates of a homogeneous Cu-5 at.% Sn alloy, which were annealed at 800°C in purified hydrogen. Porous zones were found with GB hole channels perpendicular to the surface and practically equidistant from one another. The porous zone propagation and the average pore size growth at early stages of annealing obey a parabolic law. The observed processes are caused by nucleation and growth of the Cu₃Sn phase at the free surface. The new phase works as diffusion pump pumping out Sn atoms from the alloy towards the growing compound layer. The GB channel formation has been described as a relaxation process accompanying GB interdiffusion of Sn and Cu atoms with unequal partial diffusion coefficients (D _Sn>D _Cu). Excessive vacancies appearing at the GBs due to the inequality between D _Sn and D _Cu are absorbed by bulk and GB sinks, and tensile stresses appear near the GBs stimulating hole channels or groove formation.     

Atomistic Modeling of Point Defects and Diffusion in Copper Grain Boundaries

The atomic structure of several symmetrical tilt grain boundaries (GBs) in Cu and their interaction with vacancies and interstitials as well as self-diffusion are studied by molecular statics, molecular dynamics, kinetic Monte Carlo (KMC), and other atomistic simulation methods. Point defect formation energy in the GBs is on average lower than in the lattice but variations from site to site within the GB core are very significant. The formation energies of vacancies and interstitials are close to one another, which makes the defects equally important for GB diffusion. Vacancies show interesting effects such as delocalization and instability at certain GB sites. They move in GBs by simple vacancy-atom exchanges or by long jumps involving several atoms. Interstitial atoms can occupy relatively open positions between atoms, form split dumbbell configurations, or form highly delocalized displacement zones. They diffuse by direct jumps or by the indirect mechanism involving a collective displacement of several atoms. Diffusion coefficients in the GBs have been calculated by KMC simulations using defect jump rates determined within the transition state theory. GB diffusion can be dominated by vacancies or interstitials, depending on the GB structure. The diffusion anisotropy also depends on the GB structure, with diffusion along the tilt axis being either faster or slower than diffusion normal to the tilt axis. In agreement with Borisov's correlation, the activation energy of GB diffusion tends to decrease with the GB energy.     

Grain boundary segregation diagrams of α-iron

Based on thermodynamic analysis of interfacial segregation, the segregation enthalpy H ^o of a solute I in a given matrix was found to depend linearly on two mutually independent terms reflecting the type of interface and the solid solubility limit X _infI ^sup* at temperature T and can be written as In this equation, the structural dependence of interfacial segregation is contained in H ^*() which corresponds to the extrapolated segregation enthalpy of a solute with unlimited solubility in the matrix. The product [Tln(X _infI ^sup* )] is essentially constant with temperature, and can therefore be obtained from data for maximum solid solubility, [Tln(X _infI ^sup* )]_max. The parameter v>0 represents the relationship between the activity a _infI ^sup* of a solute at the bulk solid solubility limit in a given matrix and X _infI ^sup* , a _infI ^sup* =(X _infI ^sup* ) ^v , and is characteristic for the matrix. Using recent experimental data for silicon, phosphorus, and carbon segregation at well-characterized grain boundaries in oriented bicrystals of -iron, the averaged value was determined. Values of H ^*() range from -8 kJ/mol (general grain boundaries) up to +8 kJ/mol (special grain boundaries). These values are discussed and used for a more precise and generalized construction of grain boundary segregation diagrams of -iron.     

Monte Carlo Simulation of the Concentration Dependence of Segregation at Vicinal Grain Boundaries

Using the Metropolis algorithm Monte Carlo technique solute-atom segregation is studied at two vicinal grain boundaries (GBs)—the = 5/(002)/ = 36.89° symmetrical twist and the = 5/(310)/ = 53.13° symmetrical tilt—at 850 K on the Ni-rich side of the Ni-Pt phase diagram, over the concentration range 0–10 at.% Pt. Unlike the Pt-rich side of the phase diagram the structures of both GBs remain stable in this concentration range. The dilute limit behavior for most GB sites extends to at least 0.1 at. %. At higher concentrations the effective segregation energies steadily decrease with increasing solute concentrations, due to solute-solute interactions between segregated atoms, until saturation occurs. It is argued that simple statistical mechanical models, e.g., the Fowler-Guggenheim model do not work well, even in the case of simple vicinal GBs.     

SPA-LEED studies of growth of Ag on Ag(111) at low temperatures

The surface morphology after deposition of Ag on Ag(111) at low temperatures (130–200 K) has been studied in detail with SPA-LEED (Spot-Profile Analysis of Low-Energy Electron Diffraction). The surface roughness and the mean terrace size have been quantitatively determined under various conditions. At 130 K the surface roughness increases with coverage exactly according to the relation = ^1/2, which indicates that the inter-layer diffusion can be neglected at 130 K. Although the mean terrace length decreases with increasing coverage (following an approximate power law of ^–2/3) for all studied coverages, it is much larger than expected for a pure random or Poisson-growth mode without any diffusion of the adatoms. Therefore, Ag grows on Ag(111) at this temperature without interlayer diffusion but with intra-layer diffusion. The intralayer diffusion barrierE _d has been determined by measuring the temperature dependence of the two-dimensional island density according to the nucleation theory (supposing a critical nucleus size of one). The obtained valueE _c=0.18 eV agrees with the theoretical calculations and previous measurements. Furthermore, from comparing measured and Monte-Carlo-simulated (MC) surface roughness at different deposition temperatures we obtain E=0.05 eV as a lower limit for the additional barrier at steps.     

Thermodynamics of Vacancies and Impurities at Grain Boundaries

The thermodynamics of vacancy and impurity adsorption at interfaces and grain boundaries (GBs) in solids is considered. Theoretical expressions are derived for the GB/interface free energy change caused by various levels of vacancy or impurity adsorption. This information is used to predict the behavior of vacancies at interfaces and GBs in a stress gradient and to forecast the effect of impurities on GB fracture strength. The latter predictions provide an interpretation of intergranular fracture behavior in terms of impurity adsorption and GB structural parameters such as GB width and value.     

Grain Boundary Grooving as an Indicator of Grain Boundary Phase Transformations

The atomic force microscopy (AFM) was used to study the grain boundary (GB) groove profiles far away from the melting temperature T _m. It is shown that AFM allows one to measure the temperature dependence of the GB energy in a rather broad temperature interval (from 0.85 T _m to T _m). The GB energy and GB segregation of Bi were measured at 1123 K in the interval of the Bi bulk concentration x ^v _Bi from 5 to 140 ppm Bi. The transition from monolayer to multilayer adsorption is observed for the 19a GB at 1123 K and x ^v _Bi = 60 at. ppm Bi. At the same point (1123 K and x ^v _Bi = 60 at. ppm Bi) a discontinuity of the first derivative of the GB energy is observed. These features were explained using the model of GB prewetting phase transformation developed previously.     

Tracer Diffusion of 63Ni in Nano-γ-FeNi Produced by Powder Metallurgical Method: Systematic Investigations in the C, B, and A Diffusion Regimes

⁶³Ni radiotracer diffusion in a well-compacted nanocrystalline (grain size d 80 to 100 nm) -Fe–40wt%Ni alloy was measured by the serial sectioning technique in an extended temperature range from about 610 to 1010 K. Since the material microstructure reveals two different length scales with nano-size grains forming micrometer-size clusters (or agglomerates), three main diffusion paths determine the diffusion behavior: the nanocrystalline grain interior, the nanocrystalline grain boundaries (GB), and the inter-agglomerate interfaces. The systematics of diffusion in a compacted nanostructured material with such a bimodal distribution of interface characteristics was elaborated and the experimental data were analyzed in dependence on the diffusion regime. The absolute values and the Arrhenius parameters of Ni GB diffusion in the nano--Fe–40wt%Ni alloy (D ₀ = 9.3 × 10^–4 m² s^–1 and Q = 177 kJ/mol) are similar to the Ni GB diffusivity in coarse-grained poly-crystalline -Fe. Accordingly, the nanocrystalline GBs are concluded to have quasi-equilibrium structures, particularly because of a pronounced grain growth (from about 30 to about 100 nm) during the production stage of the nanomaterial. In contrast, the inter-agglomerate interfaces, which present the fastest diffusion path in the present investigation (D ₀ = 1.9 × 10^–3 m² s^–1 and Q = 134 kJ/mol), are likely to be in a non-equilibrium state due to specific features of the applied powder metallurgical process.     

Free carrier scattering mechanism in Bi2Se3 crystals

The single valley model proposed byKöhler andLandwehr is used for the calculation of transport parameters corresponding to pure Bi₂Se₃ crystals. From the measurements of transmission, reflectivity and the Hall constant the effective masses are determined,m _c =0·13m ₀ andm _c=0·56m ₀; using a simplification described in the paper the Fermi level is calculated to lie 0·14 eV above the bottom of the conduction band. This result allows us to conclude that a mixed mechanism of free carrier scattering exists inn-Bi₂Se₃ crystals, viz., the scattering by acoustic phonons prevails contributing about 75% and the scattering on ionized impurities contributes the remaining 25%. This result applies to pure Bi₂Se₃ crystals with free electron concentration 2×10¹⁹cm^–3.     

Dynamic structure factor in a random diffusion model

Let {X _t:0} denote random walk in the random waiting time model, i.e., simple random walk with jump ratew ^–1(X _t), where {w(x):x^d} is an i.i.d. random field. We show that (under some mild conditions) theintermediate scattering function F(q,t)=E ₀ (q^d) is completely monotonic int (E ₀ denotes double expectation w.r.t. walk and field). We also show that thedynamic structure factor S(q, w)=2 ₀ cos(t)F(q, t) exists for 0 and is strictly positive. Ind=1, 2 it diverges as 1/||^1/2, resp. –ln(||), in the limit 0; ind3 its limit value is strictly larger than expected from hydrodynamics. This and further results support the conclusion that the hydrodynamic region is limited to smallq and small such that ||D |q|², whereD is the diffusion constant.     

Schrödinger Operators with Empty Singularly Continuous Spectra

Let H be a semibounded perturbation of the Laplacian H ₀ in L ²( ^d ). For an admissible function sufficient conditions are given for the completeness of the scattering system (H), (H ₀). If is the exponential function and if e^{– H} is an integral operator we denote the kernel of the difference D = e^{– H} – e^{– H} ₀ by D (x, y), > 0. The singularly continuous spectrum of H is empty if^d dx ^d dy |D(x,y)| (1 + |y|²)< for some > 1. This result is applied to potential perturbations and to perturbations by imposing Dirichlet boundary conditions.     

Positron annihilation anomaly in the high-temperature superconductor YBa2Cu3O7−x

We have measured the ac susceptibility of a wire with a Nb core (1.27 mm diam.) and a Cu cladding (0.37 mm thickness) atT50 K andB0.1 mG. Due to its proximity to Nb, the Cu becomes fully superconducting. From the data we find a breakdown fieldH _b =1.2 (mG) and a coherence length =2.2T ^–1/2 (m) for the Cu, as well as a field penetration depth -34T ^1/2 (m) at the Cu/Nb interface.     

Superradiating 4P-4S-3P cascade of sodium vapour arising on the leading edge of the exciting laser pulse

Investigations of the superradiating cascade of sodium vapour 4P-4S-3P ₁ = 2.21 µm and ₂=1.14 µm) arising on the leading edge of the exciting laser pulse were carried out. The dependences of the actual delay time _D of the ₁ pulse on the population rise time of the laser-excited 4P state were measured and compared with those calculated following the existing theoretical model. The dependence of the actual delay time _D on the inverse density of excited atoms 1/N* is also presented. Analysis of this dependence revealed the influence of the Doppler dephasing and of the second, ₂, transition on the formation of the ₁ superradiance.     

On the accommodation of extrinsic dislocations in grain boundaries

The relaxation of the extrinsic grain boundary dislocation (EGBD) stresses, created by interaction of lattice dislocations with grain boundaries (GBs), is a phenomenon which plays an important role in recrystallization and high temperature deformation of materials. The kinetics of this phenomenon, controlled by GB diffusion, are relatively well established. On the contrary, the processes which operate in order the GBs return to equilibrium are still controversed in vicinal and general GBs.The decomposition in discrete products and the rapid motion of the glissile components observed by High Resolution Electron Microscopy (HREM) in symmetrical tilt GBs support the recent incorporation model of EGBD accommodation. But, until now, these observations and this model are restricted to GBs described by the Structural Unit/Grain Boundary Dislocation (SU/GBD) model. Otherwise, the Spreading phenomenon generally observed by Transmission Electron Microscopy (TEM) in vicinal and general GBs is not clearly understood.This paper is an attempt to review the different EGBD accommodation models and to raise up the question of their relevance to account for the stress relaxation in any grain boundary.     

Nontrivial extensions of a representation of the Poincaré group with mass and helicity zero by its tensorial product

Let U(a, ) be a representation of the Poincaré group with mass and helicity zero, realized in the space of C -functions with compact support on ³, without the origin. Let U ⁽²⁾(a, ) denote the tensorial product of U(a, ) by itself. We explicitly determine the cocycles of extension of U(a, ) by U ⁽²⁾(a, ) and we prove that the nontrivial cohomology is indexed by (u(),),u()D ¹ _]0,3\,.     

Renormalized transport equations for the bistable potential model

Renormalized transport equations for general Fokker-Planck systems are derived and applied to the bistable potential model. The exact equation for the expectation value x _t can be evaluated in both domains Dx _± and xD ₀ outside and between the potential minima, leading to drastic differences of the dynamics prevailing inD _± andD ₀, respectively.     

The tortuosity of occupied crossings of a box in critical percolation

We consider the length of an occupied crossing of a box of size [0,n]×[0, 3n] ^D–1 (in the short direction) in standard (Bernoulli) bond percolation on ^D at criticality. Let ¦s _n¦ be the length of the shortest such crossing. It is believed that ¦s _n¦ ^1+c in some sense for somec>0. Here we show that if the correlation length(p) satisfies (p)p _c}–p) ^– for some <1, then with a probability tending to 1, ¦s _n¦>/C ₁ n ^1/(logn)^–(1–)/. The assumption (p)C ₃(p _c–p)^– with <1 has been rigorously established^(1,2) for largeD, but cannot hold⁽³⁾ forD=2. In the latter case, let ¦l _n¦ be the length of the lowest occupied crossing of the square [0,n]². We outline a proof ofP _pc(¦l_n¦ n ^1+c)n ^– for somec, >0. We also obtain a result about the length of optimal paths in first-passage percolation.