Diffusion in an ensemble of intersecting grain boundaries forming a triple junction

Grain boundary diffusion in an ensemble of three intersecting grain boundaries forming a triple junction is described in the framework of quasi-steady Fisher’s model. Two configurations, which differ in the number of grain boundaries adjacent to the surface with a diffusant source and in the tilt angle to the surface, are considered. Analytical expressions for the diffusant concentration distribution along each grain boundary that constitutes the triple junction and for the point of the triple junction are derived with the proviso of equal diffusion fluxes at the triple point. The expressions for the diffusant concentration distribution along the grain boundaries include not only diffusion constants (grain-boundary and bulk diffusion coefficients) but also structural characteristics of the ensemble of grain boundaries (the depth of the triple junction point and the angle between the grains forming the triple junction). It is shown that, if the coefficients of grain boundary diffusion are equal for all boundaries making the ensemble and for an equilibrium angle of 120° in a polycrystal, the diffusive mass transport rate in the triple junction zone is lower than that in a single grain boundary irrespective of the configuration of grain boundaries.     

Diffusion kinetics in an ensemble of intersecting grain boundaries. The role of a size factor

The quasi-steady state Fisher model is used to analyze the efficiency of delivery of a diffusing matter to a triple junction point in an ensemble of three intersecting grain boundaries. A version of grain boundary ensemble is considered in which two grains adjacent to a surface containing a diffusion source are at a tilt angle of 30° and the third boundary is normal to the surface. This configuration of grain boundaries corresponds to a triple junction angle of 120°. It is shown that the efficiency of delivery of a matter to the triple point depends on the ratio of the diffusion length of grain boundaries adjacent to the surface containing a diffusion source to the triple junction depth and on the relation between the grain boundary diffusion coefficients of each grain boundary in the ensemble. The conditions are found under which enhanced diffusion as compared to diffusion in a single boundary takes place in an ensemble of grain boundaries.     

Multiphase binary diffusion in infinite and semi-infinite media: Part II. On the numerical calculation of the rate constants for formation of product phases

The equations describing multiphase binary diffusion derived in the preceding article are arranged in an order such that the unknowns in the equations are expressed as two series of sequential functions of the corresponding two independent variables. The conditions to be satisfied between one series of functions and the other are given, and a numerical method for solving this type of equations of two unknowns is developed. For the numerical calculation of the parabolic rate constants for formation of product phases, data of average interdiffusion coefficients, partial molal volumes, and phase boundary compositions are required for an infinite medium. For a semi-infinite medium, in addition to these data, information on the equilibrium surface composition and the ratio of mole transfers of the two components at the surface is required.     

Investigation of Boundary Shear Stress and Pollutant Detachment from Impervious Surface during Simulated Urban Storm Runoff

Pollutant detachment rates have been determined for four chloride salts during simulated urban storm runoff. Under rainfall and/or overland flow conditions, chloride mass flux was measured and related to boundary shear stress of the test surface. Washoff coefficients, presumed to depend only on pollutant characteristics, were computed based on the slopes of dimensionless mass flux versus dimensionless time plots. Washoff coefficients were found to vary among and between the chloride compounds studied. In general, higher overland flow rates produced lower boundary shear and lower washoff coefficients. The combination of simulated rainfall and overland flow resulted in an increased boundary shear and an increased washoff coefficient. An empirical washoff coefficient based on a load characteristic curve derived from an exponential washoff relationship was also computed from the runoff data and compared with the previous washoff coefficient. A linear correlation between these two washoff coefficients was observed. The magnitude of the latter coefficient under simulated rainfall was consistent with reported values obtained from field data.     

Mechanism for diffusion induced grain boundary migration

Grain boundaries are found to migrate under certain conditions when solute atoms are diffused along them. This phenomenon, termed diffusion induced grain boundary migration (DIGM), has now been found in six systems. The observed phenomenon and empirical data are used to discard certain concepts for the driving force and the mechanism. We propose a mechanism in which differences in the diffusion coefficients of the diffusing species along the grain boundary cause a self-sustaining climb of grain boundary dislocations and motion of their associated grain boundary steps.     

Determination of sedimentation coefficients for small peptides

Direct fitting of sedimentation velocity data with numerical solutions of the Lamm equations has been exploited to obtain sedimentation coefficients for single solutes under conditions where solvent and solution plateaus are either not available or are transient. The calculated evolution was initialized with the first experimental scan and nonlinear regression was employed to obtain best-fit values for the sedimentation and diffusion coefficients. General properties of the Lamm equations as data analysis tools were examined. This method was applied to study a set of small peptides containing amphipathic heptad repeats with the general structure Ac-YS-(AKEAAKE)nGAR-NH2, n = 2, 3, or 4. Sedimentation velocity analysis indicated single sedimenting species with sedimentation coefficients (s(20,w) values) of 0.37, 0.45, and 0.52 S, respectively, in good agreement with sedimentation coefficients predicted by hydrodynamic theory. The described approach can be applied to synthetic boundary and conventional loading experiments, and can be extended to analyze sedimentation data for both large and small macromolecules in order to define shape, heterogeneity, and state of association.     

Non-invasive measurements of breast tissue optical properties using frequency-domain photon migration

A multiwavelength, high bandwidth (1 GHz) frequency-domain photon migration (FDPM) instrument has been developed for quantitative, non-invasive measurements of tissue optical and physiological properties. The instrument produces 300 kHz to 1 GHz photon density waves (PDWs) in optically turbid media using a network analyser, an avalanche photodiode detector and four amplitude-modulated diode lasers (674 nm, 811 nm, 849 nm, and 956 nm). The frequency of PDW phase and amplitude is measured and compared to analytically derived model functions in order to calculate absorption, mu a, and reduced scattering, mu s, parameters. The wavelength-dependence of absorption is used to determine tissue haemoglobin concentration (total, oxy- and deoxy- forms), oxygen saturation and water concentration. We present preliminary results of non-invasive FDPM measurements obtained from normal and tumour-containing human breast tissue. Our data clearly demonstrate that physiological changes caused by the presence of small (about 1 cm diameter) palpable lesions can be detected using a handheld FDPM probe.     

Faceting transformation and energy of a Σ3 grain boundary in silver

Grain boundary facets forming at the intersection between a grain boundary and the free surface in diffusion bonded Σ3〈011〉Ag bicrystals during prologed annealing have been characterized crystallographically by metallographic methods. It is shown that the observed faceting has qualitatively the same character as that in Σ3〈011〉 grain boundaries in Cu. The energy of an incoherent Σ3 grain boundary in Ag (210 mJ/m²) is determined from the dihedral angle of the thermal groove and the extrapolated literature data on the surface tension of Ag. The facet geometry is discussed with respect to computer simulation data on the inclination dependence of the energy of Σ3 grain boundaries in Cu. The geometrical stability of a grain boundary near the free surface is considered.     

Simplified Numerical and Analytical Approach for Solutes in Turbulent Flow Reacting with Smooth Pipe Walls

A large group of reactions that affect water quality in distribution networks occur on the pipe wall surface. Existing simulation models are usually based on cross-sectionally averaged variables that use mass-transfer coefficients derived for constant-concentration (Dirichlet) boundary conditions to account for cross-sectional variations. In the case of a first-order wall-demand problem, the boundary condition is however of Robin type. We derive a simple one-dimensional (1D) model for the radial concentration profile of a solute of arbitrary Schmidt number (Sc) reacting with pipe walls in a fully developed turbulent flow. A modified van Driest mixing length model was used to approximate the Reynolds-averaged velocity and eddy diffusivity. Numerical solutions of the 1D model agree well with a two-dimensional mass transport model and experimental data. An asymptotic solution for high Sc is derived, which is in excellent agreement with the 1D model for Sc>100. A comparison with the mass-transfer coefficients for constant-concentration boundary conditions shows that the differences between the two boundary conditions are small.     

Multicomponent diffusion simulation based on finite elements

A numerical model is presented to treat multicomponent, multiphase diffusion problems. Unlike other recent approaches that are based on the finite-difference method, analytical solutions, or particular thermodynamic models, a general procedure based on the finite-element technique is applied. The suggested formalism is based on the solution of the integral statement of the generalized diffusion equation. This treatment allows for a simple implementation of particular boundary conditions and can easily be extended from a one- to a multidimensional analysis. A brief overview of the formal representation of multicomponent diffusion coefficients, as suggested by Andersson and Agren, is given. The finite-element diffusivity matrices are evaluated for a one-dimensional bar and a two-dimensional triangular element. The model is applied to some classical examples in diffusion simulation in both one and two dimensions. The results are compared to available analytical solutions or experimental data.     

Numerical calculation of concentration distribution and dimensional change for constant-rate diffusion of a solute into a Solvent’s surface

Modified coordinate systems, such as have been utilized in the determination of mutual diffusion coefficients where volume changes occur, have been utilized in solving for concentration distributions and dimensional changes resulting from diffusion of a solute into a substrate solvent’s surface. Developed equations have been solved, utilizing simple explicit finite difference methods, for two example cases. In both examples the diffusion coefficient was assumed a function of concentration, surface flux was assumed constant, and temperature was assumed constant. Aluminum was assumed to be diffused into a copper substrate. The ξ _Y coordinate, containing an equal number of mols of the substrate component per increment, was found to be convenient for the case of a flat plate or cylindrical substrate. Results have been presented graphically. This treatment is applicable to single-phase systems wherein the arrival rate of solute atoms at the substrate surface, either by ionic transport in high-temperature electrodiffusion cells or from the vapor phase, is equal to the diffusion rate into the surface.     

Scattering of SH Waves by Truncated Semicircular Canyon

The judicious region-matching technique is adopted to derive a series solution for shear horizontal waves scattering from a truncated semicircular canyon on the ground surface. A circular-arc auxiliary boundary is introduced to divide the analyzed region into two subregions. The antiplane motion of each subregion is represented in terms of an infinite series of cylindrical wavefunctions with unknown coefficients. By employing the Graf’s addition formula, boundary conditions on the curved canyon surface and continuity conditions on the auxiliary boundary, the unknown coefficients can be determined. Two modifications of the present theoretical derivation are done for the semicircular canyon and free surface cases. The plotted results reveal how the surface displacement amplitudes are influenced by varying the truncation depth.     

Surface effects and flow conditions in small volume melts with varying sulphur content

The surface effects and flow behaviour in melts with a volume of a few cubic millimetres were investigated using 1 mm thick X5CrNi18‐9 (AISI 304) stainless steel strips after partial remelting by a TIG (tungsten inert gas) burner. In the industrially produced material the S mass content was varied between 20 and 100 ppm. The surface excess of sulphur indicates that the surface saturation was not achieved compared to the values in the Fe‐S system. The thickness of the layer with high sulphur enrichment depends on the S activity and varied between approximately 260 and 460 nm and achieves the bulk S content at a distance of approximately 1.2 μm from the surface. This layer is noticeably thinner than the diffusion boundary layer and considerably thinner than the flow boundary layer. The topography of the rapidly solidified liquid pool makes it possible, taking previously estimated temperature coefficients of surface tension into account, to explain the flow behaviour in the melt with regard to the S content and to the temperature gradients. The results obtained indicate that the shearing force caused by surface tension gradients (Marangoni effect) controls the flow conditions in small volume melts.     

Shrinkage and splitting of microcracks under pressure simulated by the finite-element method

The two-dimensional finite-element method is applied to analyze the shrinkage and splitting of microcracks regularly arranged on or perpendicular to a grain boundary under pressure. Grain-boundary and surface diffusions are coupled by the boundary conditions at the triple point of the microcrack surface and the grain boundary. The shrinkage and splitting processes for the two kinds of microcracks are revealed by detailed finite-element analyses. For the microcrack lying on a grain boundary, it first shrinks to a small void shape, then the void is split by the grain boundary and the two split voids assume a cylindrical shape under the capillary force of the surface. For the microcrack perpendicular to the grain boundary, it is split into two segments by the grain boundary during the early stage of shrinkage. Then, the split microcracks stop shrinking and evolve into two cylindrical channels with a circular section by the capillary force of the surface. These evolution processes are controlled by the applied pressure, microcrack spacing, ratio of grain-boundary diffusion to surface diffusion, and equilibrium dihedral angle, defined by surface and grain-boundary tensions. The influences of these controlled parameters on the evolution processes are numerically clarified based on a great number of finite-element analyses.     

A numerical simulation of the D.C. continuous casting process including nucleate boiling heat transfer

The steady-state thermal problem associated with the direct-chill continuous casting of A6063 aluminum cylindrical ingots is solved using the numerical finite element technique. Excellent correlation is demonstrated between the numerical model and experimental data from ingots cast at two different speeds. By application of the model, effective heat transfer coefficients are calculated as a function of vertical position on the outside surface of the ingot. It is shown that direct application of these coefficients to the modeling of different casting situations will produce substantial errors in the region in which heat transfer is by nucleate boiling. Using theories of nucleate boiling with forced convection and film cooling, a method is developed to calculate the external boundary conditions in the submold region of the ingot, thus making it possible for the first time to define explicitly all of the thermal boundary conditions associated with this casting configuration. These theories are incorporated into the numerical model, and a subsequent simulation shows excellent agreement with experimental data from a third ingot.     

High-rate carburizing in a glow- discharge methane plasma

Carburizing at high temperature (1040°C) in a glow-discharge methane plasma signifi-cantly reduces carburizing time. Sufficient carbon to produce a 1.0 mm case on conven-tional carburizing steels can be introduced in 10 min at methane pressures in the range of 1.3 to 2.7 kPa (10 to 20 Torr). To reduce the carbon concentration at the surface to an acceptable level (007E1.0 wt pct) the plasma carburizing stage is followed by a short (007E30 min) diffusion step. To aid in optimizing the division of time between the carburizing and diffusion stages, and in separating the effects of plasma carburizing and high-temperature diffusion, a computer program was written to model the diffusion equation for appropriate initial and boundary conditions. Both model and experimental results show that a diffusion time/carburizing time ratio of at least 3: 1 is required. The diffusion model indicates that the exceptionally high carburizing rates observed arise from the rapid infusion of carbon into the surface from the plasma, during the carburizing stage and not from anomalously high diffusion rates. Exceptionally uniform cases on surfaces of irregular geometry are achieved, and significant amounts of natural gas are saved owing to both the reduced car-burizing time and the low pressures employed.     

Application of fractal theory on raceway boundary in COREX melter–gasifier model

Abstract

The raceway has been studied extensively both theoretically and experimentally. In this study, particle velocity contours have been developed to define the raceway boundary. The raceway boundary is coarse and fragmentised, but all of the previous studies are based on Euclidean geometry, which regard the dimension of the raceway as an integer. In this paper, the fractal method of calculating raceway size, which describes the boundary with extremely irregular or fragmented characteristics, is on the data from a cold physical model of the raceway. The results show that the precise raceway boundary can be obtained by the particle velocity contours, that the surface area of the ellipsoidal raceway based on fractal theory is larger than that based on Euclidean geometry and that the data can be used as original boundary conditions of the flow and chemical reaction in the raceway region.     

A numerical study of cavity growth controlled by coupled surface and grain boundary diffusion

The diffusive growth of both two dimensional and axisymmetric cavities initially having equilibrium shapes and located on grain boundaries loaded in tension is studied using finite difference techniques. The shape evolution and growth kinetics of individual cavities as well as the time required for adjacent cavities to grow together is studied as a function of applied stress and the ratio of grain boundary to surface diffusivity. A key feature of this treatment is that the diffusional processes in the grain boundary and on the cavity surface are coupled by boundary conditions at the tip of the cavity. When surface diffusion is much slower than grain boundary diffusion, the cavities become crack-like during growth, and the fracture time varies reciprocally with the third power of the applied stress. When grain boundary diffusion is the slower process, the cavities remain rounded during growth, and the fracture time varies reciprocally with the first power of the stress. The transition between these limiting kinds of behavior is described and the results are compared with previous treatments of these problems.     

Elastic fields about a perturbed cylindrical inclusion

The elastic fields about an inclusion having a shape that is slightly perturbed from that of a circular cylinder are calculated. Both the inclusion and the surrounding matrix are assumed to be isotropic linear elastic materials. Two sets of boundary conditions are applied at the inclusion matrix interface. The first set, corresponding to a coherent interface, consists of balance offorces and continuity of displacement at the interface. The second set, corresponding to a “greased” or incoherent interface, consists of balance of forces, continuity of the component of displacement normal to the interface, and vanishing tangential forces at the interface. Differences between the elastic fields calculated for these two sets of boundary conditions are manifested only in terms which are first order in the amplitude of the shape perturbation. For coherent boundary conditions, the trace of the stress tensor is zero in the matrix and constant in the inclusion when the matrix and inclusion have equal shear moduli. For “greased” boundary conditions, the trace of the stress tensor is generally not constant in either the matrix or the inclusion, even when their shear moduli are equal. The importance of the perturbed elastic fields in relation to morphological stability theory is illustrated by using a constitutive relation for diffusion that depends, in part, on the gradient of the trace of the stress tensor.     

Predicting Interfacial Diffusion Coefficients for Fluxes across the Sediment-Water Interface

Calculations of interfacial diffusion coefficients are often based on tracer flux or penetration into permeable media or sediments. Based on previous investigations, a new empirical relationship for tracer-based interfacial diffusion coefficients is derived. This relationship is a powerful tool for estimating interfacial fluxes over a range of environmentally relevant conditions. Support for this relationship was found in experiments targeting the slip velocity at porous media–boundary layer interfaces. Slip measurements from flume experiments using flat permeable sediment beds and from previous studies using high permeability media were converted to interfacial diffusion coefficients based on the momentum flux needed to drive the observed interstitial flow. Slip-based estimates compared well with the tracer-based predictions over the entire range of flow-permeability conditions. This study presents a relationship for predicting fluxes across the sediment-water interface that is driven by permeability-scale processes. Predicted fluxes are comparable to those documented for other processes driving interfacial transport, such as bed topography and sediment transport.