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.     

Perturbation theory of scattering from irregular bodies

A perturbation solution is derived for the following problem: A time harmonic wave of amplitude ψ, propagating in a medium with wave number k, is incident on an irregular volume V, inside of which the propagation constant k′(r) can be an arbitrary function of | r |, where r is a position vector with origin inside V. The boundary conditions are that both ψ and its normal derivative ∂ψ/∂n may be discontinuous across the surface of V. Special cases of these conditions correspond to acoustic scattering, to B-wave scattering from a dielectric cylinder, or to the classical Dirichlet (ψ = 0) or Neumann (∂ψ/∂n = 0) surface conditions. An integral equation is derived that satisfies the appropriate differential equations both outside and inside the body, and satisfies the boundary conditions as well. This equation is reduced to a set of linear algebraic equations by expansion in a certain basis set and these linear equations are then solved in a perturbation approximation for the case that the surface of the body differs from a sphere or cylinder by a small parameter λ. Comparison is made with formulae in the literature, and except for some minor discrepancies, which are here corrected, there is general agreement.     

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.     

Diffusion through a polycrystalline thin film

Analytical solutions to Fick’s second law of diffusion have been simultaneously derived without the restrictions of parabolic profiles along the x-axis in grain boundaries and expressed in a series for both grain interior and grain boundary diffusing through a polycrystalline thin film. The analysis takes segregation of diffusion species at grain boundaries into account. The analytic solutions lead to the concentration profiles in the grain interior and in the grain boundary, to the average-integrated amount of diffusion species at the exit surface, and to the time lag, which can be technologically used for depth profile studies and kinetic accumulation measurements.     

Volume and grain boundary diffusion of implanted 113Sn in aluminium

Volume and grain boundary diffusion of ¹¹³Sn in aluminium was investigated with the radiotracer method. The implantation technique was used for tracer deposition to avoid problems of tracer hold-up caused by the oxide layer always present on aluminium. The diffusion penetration was chosen large enough to permit serial sectioning of samples with the aid of a microtome.The temperature dependence of the volume diffusivity was determined as D(T)=4.54×10^–5×exp[–(114.5±1.2)kJmol^–1/RT] m ² s ^–1. This confirms previous measurements from our group which already showed that Sn is the fastest foreign metal diffusor so far investigated in aluminium.Grain boundary diffusion of ¹¹³Sn in Al polycrystals was measured in the type-B kinetic regime. The grain boundary diffusion product P=sD _gb (s=segregation factor, =grain boundary width, D _gb=grain boundary diffusivity) was found to be strongly affected by the impurity content of aluminium. For Al polycrystals of 99.9992% nominal purity we obtained P _5N(T)=1.08×10^–8exp [–(96.9±7.5) kJ mol^–1/RT] m³ s^–1 and for less pure Al polycrystals of 99.99% nominal purity P _4N(T)=3.0×10^–10 exp [–(90.1±4.2) kJ mol^–1/RT] m³ s^–1 was determined. The grain boundary diffusion product in the purer material is more than one order of magnitude higher than in the less pure material. Very likely this is an effect of co-segregation of non-diffusant impurities into the grain boundaries.     

Positron lifetime as a function of grain size

Published data on positron annihilation lifetime in copper as a function of grain size have been analyzed to show that there is a linear relationship between the internal grain boundary surface area, per unit volume,S _v, and the positron lifetime, τ. The analysis indicates that grain boundaries are important in the trapping of positrons. It is suggested that the slope of the resulting straight line,dS _v/dτ, can be used to determine the annihilation rate of the grain boundaries.     

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.     

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:

Guidelines for choosing the transition matrix in Monte Carlo methods using Markov chains

The sampling method proposed by Metropolis et al. (J. Chem. Phys. 21 (1953), 1087) requires the simulation of a Markov chain with a specified π as its stationary distribution. Hastings (Biometrika 57 (1970). 97) outlined a general procedure for constructing and simulating such a Markov chain. The matrix P = {p_ij} of transition probabilities is constructed using a defined symmetric function s and an arbitrary transition matrix Q. With respect to asymptotic variance reduction, Peskun (Biometrika 60 (1973), 607) determined, for a given Q, the optimum choice for s_ij. Here, guidelines are given for choosing Q so that the resulting Markov chain sampling method is as precise as is practically possible. Examples illustrating the use of the guidelines, including potential applications to problems in statistical mechanics and to the problem of estimating the probability of an simple event by “hit-ormiss” Monte Carlo in conjunction with Markov chain sampling, are discussed.     

Orientational dynamics of superfluid He: A “two-fluid” model. II. Orbital dynamics

We present a phenomenological theory of the homogeneous orbital dynamics of the class of “separable” anisotropic superfluid phases which includes the ABM state generally identified with ³He-A. The theory is developed by analogy with the spin dynamics described in the first paper of this series; the basic variables are the orientation of the Cooper-pair wavefunction (in the ABM phase, the l-vector) and a quantity K which we visualize as the “pseudo-angular momentum” of the Cooper pairs but which must be distinguished, in general, from the total orbital angular momentum of the system. In the ABM case l is the analog of d in the spin dynamics and K of the “superfluid spin” S_p. Important points of difference from the spin case which are taken into account include the fact that a rotation of l without a simultaneous rotation of the normal-component distribution strongly increases the energy of the system (“normal locking”), and that the equilibrium value of K is zero even for finite total angular momentum. The theory does not claim to handle correctly effects associated with any intrinsic angular momentum arising from particle-hole asymmetry, but it is shown that the magnitude of this quantity can be estimated directly from experimental data and is extremely small; also, the Landau damping does not emerge automatically from the theory, but can be put in in an ad hoc way. With these provisos the theory should be valid for all frequencies irrespective of the value of ωτ. (Δ = gap parameter, τ = quasi-particle relaxation time.) It disagrees with all existing phenomenological theories of comparable generality, although the disagreement with that of Volovik and Mineev is confined to the “gapless” region very close to T_c.The phenomenological equations of motion, which are similar in general form to those of the spin dynamics with damping, involve an “orbital susceptibility of the Cooper pairs” χ_orb(T). We give a possible microscopic definition of the variable K and use it to calculate χ_orb(T) for a general phase of the “separable” type. The theory is checked by inserting the resulting formula in the phenomenological equations for ωτ 1 and comparing with the results of a fully microscopic calculation based on the collisionless kinetic equation; precise agreement is obtained for both the ABM and the (real) polar phase, showing that the complex nature of the ABM phase and the associated “pair angular momentum” is largely irrelevant to its orbital dynamics. We note also that the phenomenological theory gives a good qualitative picture even when ω Δ(T), e.g., for the flapping mode near T_c. Our theory permits a simple and unified calculation of (1) the Cross-Anderson viscous torque in the overdamped regime, (2) the flapping-mode frequency near zero temperature, (3) orbital effects on the NMR, both at low temperatures and near T_c, (4) the orbit wave spectrum at zero temperature (this requires a generalization to inhomogeneous situations which is possible at T = 0 but probably not elsewhere). We also discuss the possibility of experiments of the Einstein-de Haas type. Generally speaking, our results for any one particular application can be also obtained from some alternative theory, but in the case of orbital and spin relaxation very close to T_c (within the “gapless” region) our predictions, while somewhat tentative and qualitative, appear to disagree with those of all existing theories. We discuss briefly how our approach could be extended to apply to more general phases.     

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)     

Condition for adiabatic passage in the earth’s-field NMR technique

The equation of motion dM/dt=γ M×B(t) is solved for the case B(t)=jB_p(t)+kB_e. The field B_e is a small static field, typically the earth’s field. The field B_p(t) decays exponentially toward zero with time constant T. This decay is produced by an overdamped switching transient that occurs near the end of the rapid cutoff of the coil current used to polarize the sample. It is assumed that B_p is initially large compared to B_e, and that magnetization M is initially along the resultant field B. Exact solutions are obtained numerically for several decay time constants of B_p, and the motion of M is depicted graphically. It is found that for adiabatic passage, the final cone angle β of the precession in field B_e is related to the decay time constant of B_p by β=2e^{−(π/2)ω_eT}. This is confirmed by measurements of the amplitudes of the ensuing free-precession signals for various decay rates of B_p. Near-perfect adiabatic passage (magnetization aligned within 2° of the earth’s field) can be achieved for time constants T2.6/ω_e. For the case of sudden passage, an approximate analytic solution is developed by linearizing the equation of motion in the laboratory frame of reference. For the adiabatic case, an approximate analytic solution is obtained by linearizing the equation of motion in a rotating frame of reference that follows the resultant field B=B_p+B_e.     

Accurate, high-order representation of complex three-dimensional surfaces via Fourier continuation analysis

We present a new method for construction of high-order parametrizations of surfaces: starting from point clouds, the method we propose can be used to produce full surface parametrizations (by sets of local charts, each one representing a large surface patch – which, typically, contains thousands of the points in the original point-cloud) for complex surfaces of scientific and engineering relevance. The proposed approach accurately renders both smooth and non-smooth portions of a surface: it yields super-algebraically convergent Fourier series approximations to a given surface up to and including all points of geometric singularity, such as corners, edges, conical points, etc. In view of their C^∞ smoothness (except at true geometric singularities) and their properties of high-order approximation, the surfaces produced by this method are suitable for use in conjunction with high-order numerical methods for boundary value problems in domains with complex boundaries, including PDE solvers, integral equation solvers, etc. Our approach is based on a very simple concept: use of Fourier analysis to continue smooth portions of a piecewise smooth function into new functions which, defined on larger domains, are both smooth and periodic. The “continuation functions” arising from a function f converge super-algebraically to f in its domain of definition as discretizations are refined. We demonstrate the capabilities of the proposed approach for a number of surfaces of engineering relevance.     

Solution of Time-Dependent Diffusion Equations with Variable Coefficients Using Multiwavelets

A new numerical algorithm is developed for the solution of time-dependent differential equations of diffusion type. It allows for an accurate and efficient treatment of multidimensional problems with variable coefficients, nonlinearities, and general boundary conditions. For space discretization we use the multiwavelet bases introduced by Alpert (1993,SIAM J. Math. Anal.24, 246–262), and then applied to the representation of differential operators and functions of operators presented by Alpert, Beylkin, and Vozovoi (Representation of operators in the multiwavelet basis, in preparation). An important advantage of multiwavelet basis functions is the fact that they are supported only on non-overlapping subdomains. Thus multiwavelet bases are attractive for solving problems in finite (non periodic) domains. Boundary conditions are imposed with a penalty technique of Hesthaven and Gottlieb (1996,SIAM J. Sci. Comput., 579–612) which can be used to impose rather general boundary conditions. The penalty approach was extended to a procedure for ensuring the continuity of the solution and its first derivative across interior boundaries between neighboring subdomains while time stepping the solution of a time dependent problem. This penalty procedure on the interfaces allows for a simplification and sparsification of the representation of differential operators by discarding the elements responsible for interactions between neighboring subdomains. Consequently the matrices representing the differential operators (on the finest scale) have block-diagonal structure. For a fixed order of multiwavelets (i.e., a fixed number of vanishing moments) the computational complexity of the present algorithm is proportional to the number of subdomains. The time discretization method of Beylkin, Keiser, and Vozovoi (1998, PAM Report 347) is used in view of its favorable stability properties. Numerical results are presented for evolution equations with variable coefficients in one and two dimensions.     

Molecular dynamics simulation of stress and grain evolution

A three-dimensional molecular dynamics simulation is carried out to study the evolution of grains and stresses during the deposition of atoms on the (100) plane of a fcc regular crystal, using the cubic system with x–y periodic boundary conditions. At the bottom an atomic surface and at the top a reflecting wall are assumed. Atoms in the system interact via the Lennard–Jones potential. During simulation the films grow according to the Volmer–Weber mode and exhibit specific shape of the stress curves. When the film becomes continuous, the stress during the growth possesses a maximum value, but later new grain boundaries are formed. Individual atoms in the grain boundaries generate compressive stress in the films.     

Grain Boundary Grooving Accelerated by Local Plasticity as a Possible Mechanism of Liquid Metal Embrittlement

Kinetic mechanism of premature grain boundary (GB) failure under simultaneous action of tensile stress and wetting liquid metals (LME) is not clear yet. We summarize briefly the stress intensity, surface energy and GB segregation effects in LME crack kinetics and the explanations that have been put forward. A possible mechanism of LME is outlined. It is proposed that GB cracks under LME propagate by Mullins grooving controlled by bulk diffusion in liquid and dramatically accelerated by periodic blunting of the groove tip by local plastic flow. It is found that this GALOP (Grooving Accelerated by Local Plasticity) mechanism captures the major experimental observations of stress intensity, surface energy and GB segregation effects in LME fairly well. A parallel between the GALOP mechanism and the GB subcritical crack growth under creep in inert atmosphere is mentioned.     

Anomalous effects of hydrostatic pressure on ice surface self-diffusion

This paper presents an experimental study of the formation time of the groove that occurs at the intersection of a grain boundary and the free surface of a pure ice sample immersed in silicone oil at a pressure of approximately 10 MPa and temperature higher than ? 25 °C. Using the theoretical approach presented by Mullins (1957) [14], the surface diffusion coefficient is obtained. It is shown that, at 10 MPa, the ice surface diffusion coefficient is more than twice that observed at atmospheric pressure.     

Grain boundary engineering: fatigue fracture

Abstract

Grain boundary engineering has revealed significant enhancement of material properties by modifying the populations and connectivity of different types of grain boundaries within the polycrystals. The character and connectivity of grain boundaries in polycrystalline microstructures control the corrosion and mechanical behaviour of materials. A comprehensive review of the previous researches has been carried out to understand this philosophy. Present research thoroughly explores the effect of total strain amplitude on phase transformation, fatigue fracture features, grain size, annealing twinning, different grain connectivity and grain boundary network after strain controlled low cycle fatigue deformation of austenitic stainless steel under ambient temperature. Electron backscatter diffraction technique has been used extensively to investigate the grain boundary characteristics and morphologies. The nominal variation of strain amplitude through cyclic plastic deformation is quantitatively demonstrated completely in connection with the grain boundary microstructure and fractographic features to reveal the mechanism of fatigue fracture of polycrystalline austenite. The extent of boundary modifications has been found to be a function of the number of applied loading cycles and strain amplitudes. It is also investigated that cyclic plasticity induced martensitic transformation strongly influences grain boundary characteristics and modifications of the material’s microstructure/microtexture as a function of strain amplitudes. The experimental results presented here suggest a path to grain boundary engineering during fatigue fracture of austenite polycrystals.     

Grain Structure of Ni3Mn Alloy Ordered in Annealing

The grain boundary ensembles in Ni₃Mn alloy with L1₂ superstructure and different manganese content were investigated. It was found that the process of atomic ordering is accompanied by the migration of grain boundaries. The presence of oxygen atoms in solid solution and oxide particles at the grain bound- aries hinders the formation of the long-range atomic order in the crystalline lattice and migration of grain boundaries.