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Local and Effective Elastic Properties of Grain Boundaries in Silicon
Abstract
When considering the mechanical behaviour of materials an important property is the tensor of elastic moduli. Recently, local elastic moduli of interfaces have been defined and studied for metallic materials [1 to 3]. In these works grain boundaries are regarded as heterogeneous continua composed of ‘phases’ associated with individual atoms which possess elastic moduli identified with the atomic-level moduli evaluated at corresponding atomic positions. From this representation it is possible to define the ‘effective’ moduli of the grain boundary region. In this paper this concept is developed for materials with covalent character of bonding, specifically silicon. Using the Tersoff's potential [4, 5], the atomic-level and effective elastic moduli of the interfacial region have been evaluated for three alternate structures of the Σ = 3 (112-)/[11-0] tilt boundary. These calculations are then compared with the continuum bounds on the effective moduli evaluated using the classical minimum-energy principles of elasticity. The effective moduli calculated in the atomistic framework are generally within the continuum bounds and thus the present study demonstrates that the heterogeneous continuum model of the interfaces is appropriate for the description of the elastic properties of grain boundaries in silicon. An important aspect addressed in this study is the uniqueness of interfacial elastic moduli since their evaluation involves the energy associated with an atom which cannot be defined uniquely. The calculations have been made for two different partitions of the total energy into energies associated with individual atoms. These two partitions lead to almost identical results for the effective moduli and continuum bounds when the tensor of the atomic-level moduli is positive definite. When some atomic-level moduli are not positive definite the results may depend on the chosen energy partition.
... Wolf and co-workers Wolf, 1990), who studied superlattices of (001) twist boundaries, as well as Adams et al. (Adams, 1989), who examined the Σ5 twist boundary in a thin film of copper, have found an increase of the Young's modulus perpendicular to the boundary plane and a substantial decrease of the shear modulus in the boundary plane in the atomic layers adjacent to the boundary. Bassani and coworkers Vitek, 1994;Marinopoulos, 1998) defined the local atomic elastic modulus tensor and determined the values of the local elastic modulus tensor near grain boundaries in several face center cubic metals using molecular dynamic simulations. They also found that the local elastic moduli are significantly different for atoms near the grain boundaries. ...
... Bassani and co-workers Bassani et al., 1992;Vitek, 1994;Marinopoulos et al., 1998) defined the local elastic modulus tensor and determined the values of the local elastic modulus tensor near grain boundaries in several face center cubic metals using molecular dynamic simulations. They, too, found that the local elastic moduli are significantly different for atoms near the grain boundary. ...
... Note that, once the inter-atomic potential ( ) n E is given, the coefficients ( ) For an atom deep inside the semi-infinite crystal (or, equivalently, an atom in an infinite crystal), the local elastic stiffness tensor can be defined as Bassani et al., 1992;Marinopoulos et al., 1998), where ijkl C is the elastic stiffness tensor of the perfect crystal lattice, and the 3 3 × matrix ij D is given by ...
In this dissertation, an innovative approach combining continuum mechanics and atomistic simulations is exposed to develop a nanomechanics theory for modeling and predicting the macroscopic behavior of nanomaterials. This nanomechanics theory exhibits the simplicity of the continuum formulation while taking into account the discrete atomic structure and interaction near surfaces/interfaces. There are four primary objectives to this dissertation. First, theory of interfaces is revisited to better understand its behavior and effects on the overall behavior of nanostructures. Second, atomistic tools are provided in order to efficiently determine the properties of free surfaces and interfaces. Interface properties are reported in this work, with comparison to both theoretical and experimental characterizations of interfaces. Specifically, we report surface elastic properties of groups 10 11 transition metals as well as properties for low-CSL grain boundaries in copper. Third, we propose a continuum framework that casts the atomic level information into continuum quantities that can be used to analyze, model and simulate macroscopic behavior of nanostructured materials. In particular, we study the effects of surface free energy on the effective modulus of nano-particles, nanowires and nano-films as well as nanostructured crystalline materials and propose a general framework valid for any shape of nanostructural elements / nano-inclusions (integral forms) that characterizes the size-dependency of the elastic properties. This approach bridges the gap between discrete systems (atomic level interactions) and continuum mechanics. Finally this continuum outline is used to understand the effects of surfaces on the overall behavior of nano-size structural elements (particles, films, fibers, etc.) and nanostructured materials. More specifically we will discuss the impact of surface relaxation, surface elasticity and non-linearity of the underlying bulk on the properties nanostructured materials. Ph.D. Committee Chair: Jianmin Qu; Committee Member: David McDowell; Committee Member: Elisa Riedo; Committee Member: Min Zhou; Committee Member: Mo Li
... But there is a dramatic reduction in the resistance to shear parallel to the grain boundary plane (Kluge et al., 1990), which needs to be taken care of irrespective of the volume fraction of grain boundaries. Bassani et al. (1992); Marinopoulos et al. (1998) determined the values of the local elastic modulus tensor near grain boundaries in several face centers cubic metals using molecular dynamic simulations and observed that the Youngs modulus of a thin film of copper had increased perpendicular to the boundary plane and a substantial decrease of the shear modulus in the boundary plane in the atomic layers adjacent to the boundary (Bassani et al., 1992;Marinopoulos et al., 1998). ...
... But there is a dramatic reduction in the resistance to shear parallel to the grain boundary plane (Kluge et al., 1990), which needs to be taken care of irrespective of the volume fraction of grain boundaries. Bassani et al. (1992); Marinopoulos et al. (1998) determined the values of the local elastic modulus tensor near grain boundaries in several face centers cubic metals using molecular dynamic simulations and observed that the Youngs modulus of a thin film of copper had increased perpendicular to the boundary plane and a substantial decrease of the shear modulus in the boundary plane in the atomic layers adjacent to the boundary (Bassani et al., 1992;Marinopoulos et al., 1998). ...
The effective elastic properties of porous rock depend on the change of the surface energy or moisture content along with its porosity, the morphology of the pore space, mineralogy, grain size and state of grain contact zone, fluid contents etc. The impact of surface energy on elastic properties of rock is under-explored. We develop an FEM based numerical model which is capable of predicting effective elastic properties of rock directly from the micro-computed tomography image under different intensities of the surface energy. Dry Bentheimer sandstone exhibits 10.8% increase of bulk modulus due to the surface energy compared to 100% humid sample, which is validated with the experimentally determined impact of the surface energy. The simulated elastic moduli with dynamic input parameters are usually compared
with dynamic moduli and there is no argument against the approach. However, static tests are more representative of mechanical properties of rocks than dynamic tests and there is an obvious discrepancy between static and dynamic moduli. In this study, these are validated with experimentally determined static moduli for the first time to develop a framework to simulate static experimental procedure. Of
course, we use static input parameters for simulation. At zero confining pressure, the agreement is worse, partly due to inherent image resolution limitation. At 25 MPa confining pressure, most of the sub-micron level features are effectively closed. Due to this fact, elastic parameters estimated at this confining pressure are in excellent
agreement with simulated values. In addition, the impact of the grain contact zone on effective elastic properties is analyzed and estimated based on high-resolution SEM image, grain partition based segmented image and effective medium theory. Sister samples and a similar amount of strain amplitude are used for better accuracy of the validation process in laboratory measurements and numerical simulations.
... Wolf and coworkers Kluge et al., 1990;Wolf and Kluge, 1990), who studied superlattices of (0 0 1) twist boundaries, as well as Adams et al. (1989), who examined the S ¼ 5ð0 0 1Þ twist boundary in a thin film of copper, have found an increase of the Young's modulus perpendicular to the boundary plane and a substantial decrease of the shear modulus in the boundary plane in the atomic layers adjacent to the boundary. Bassani and co-workers Bassani et al., 1992;Vitek et al., 1994;Marinopoulos et al., 1998) defined the local elastic modulus tensor and determined the values of the local elastic modulus tensor near grain boundaries in several face center cubic metals using molecular dynamic simulations. They, too, found that the local elastic moduli are significantly different for atoms near the grain boundaries. ...
... Wolf and coworkers Kluge et al., 1990;Wolf and Kluge, 1990), who studied superlattices of (0 0 1) twist boundaries, as well as Adams et al. (1989), who examined the S ¼ 5ð0 0 1Þ twist boundary in a thin film of copper, have found an increase of the Young's modulus perpendicular to the boundary plane and a substantial decrease of the shear modulus in the boundary plane in the atomic layers adjacent to the boundary. Bassani and co-workers Bassani et al., 1992;Vitek et al., 1994;Marinopoulos et al., 1998) defined the local elastic modulus tensor and determined the values of the local elastic modulus tensor near grain boundaries in several face center cubic metals using molecular dynamic simulations. They, too, found that the local elastic moduli are significantly different for atoms near the grain boundaries. ...
Atoms at a free surface experience a different local environment than do atoms in the bulk of a material. As a result, the energy associated with these atoms will, in general, be different from that of the atoms in the bulk. The excess energy associated with surface atoms is called surface free energy. In traditional continuum mechanics, such surface free energy is typically neglected because it is associated with only a few layers of atoms near the surface and the ratio of the volume occupied by the surface atoms and the total volume of material of interest is extremely small. However, for nano-size particles, wires and films, the surface to volume ratio becomes significant, and so does the effect of surface free energy. In this paper, a framework is developed to incorporate the surface free energy into the continuum theory of mechanics. Based on this approach, it is demonstrated that the overall elastic behavior of structural elements (such as particles, wires, films) is size-dependent. Although such size-dependency is negligible for conventional structural elements, it becomes significant when at least one of the dimensions of the element shrinks to nanometers. Numerical examples are given in the paper to illustrate quantitatively the effects of surface free energy on the elastic properties of nano-size particles, wires and films.
The chapter contains sections titles:
Characteristics of plastic deformation of metals are strongly dependent on the underlying microstructure. Moreover, when dislocations are confined to move in small volumes, the yield strength and hardening rate in monotonic deformation and the fatigue strength under cyclic loading can be significantly different from corresponding conditions where dislocations are not confined. We first discuss the main mechanisms that control plastic flow, and present results of computer simulations of plasticity in small and confined volumes. This includes plastic deformation in nano-scale anisotropic multi-layered structures, where confinement is achieved through elastic modulus mismatch, plastic flow in micron-size single crystals, where confinement is attained through free surfaces, and the confined motion of dislocations in Persistent Slip Band (PSB) channels, where confinement is determined by dislocation interaction with dipolar arrays under cyclic loading conditions.
The sections in this article areIntroductionGrain Boundary Structure: Concepts and ToolsGrain Boundary DefinitionsGeometrical ConceptsDislocation ModelPrimary Dislocation NetworkSecondary Dislocation NetworkStress Field Associated with Grain BoundariesStructural Unit DescriptionsStick and Ball Structural UnitsEnergetic Structural UnitsAlgebraic Structural UnitsStructural Units and Dislocations/DisclinationsThe Limits of the Structural Unit DescriptionsComputer Simulation TechniquesMethodsBoundary ConditionsInteraction LawsExperimental TechniquesGrain Boundary Structure: Experience and Simulation ResultsSilicon and GermaniumTilt Grain BoundariesTwist Grain BoundariesDiamondSiCGaAsGaNAlNNiOComments on Grain Boundary StructuresElectrical Properties of Grain BoundariesIntroductionElectrical Effects Induced by Grain BoundariesElectronic States Associated with a Grain BoundaryPotential Barrier and Transport PropertiesDynamic Properties and Recombination PropertiesExperimental Methods for Measuring the Grain Boundary Electrical ActivityMethods Based on TransportTransient Methods
Correlation Between Electrical Activity and StructureTransport Experiments in BicrystalsTransient Properties Measured on BicrystalsEmission and Capture Properties of Silicon and Germanium Grain BoundariesPolycrystalline SiliconIntrinsic or Extrinsic Origin of Electrical Activity of Grain BoundariesImpurity Segregation and Precipitation Induced by Grain BoundariesIntroductionDopant ElementsOxygen and SulfurTransition ElementsCopperNickelIronConclusions
Mechanical Properties of Grain Boundaries in SemiconductorsIntroductionInteraction Between Dislocations and Grain BoundariesDislocation AbsorptionDislocation Transmission Across Grain BoundariesGrain Boundaries as a Dislocation SourceGrain Boundary Dislocation MovementPhysical ConsequencesGrain Boundary MigrationRecovery of the Grain Boundary Structure and CavitationDeformation ModellingConclusions
Stresses and strains around a dislocation at a grain boundary in germanium are measured by a combination of high-resolution electron microscopy and geometric phase analysis. The method is established by first measuring the strains around a matrix dislocation in silicon. Stresses are determined using linear elastic theory and bulk elastic constants. Strain measurements are shown to agree with theoretical calculations based on linear anisotropic elastic theory to 0.2% at a spatial resolution of 2-3 nm. A dislocation constricted at a coherent twin boundary in germanium is subsequently analyzed. The method is adapted to cope with the problem that the reference lattice is not identical for the whole field of view, due to the grain boundary. Strains are compared with theoretical calculations of a matrix dislocation in germanium. Whereas strains in the grains on either side of the twin boundary agree closely with the isolated dislocation case, significant additional strains are localized at the boundary plane. By comparing the stresses and strains across the boundary plane, values for the elastic modulus of the twin boundary are proposed. The significant reduction in elastic modulus for the boundary, when compared to bulk elastic constants, is interpreted in terms of the non-equilibrium configuration of the boundary. An extension of the method is proposed to measure more generally the elastic properties of grain boundaries and interfaces.
It is important to clarify the mechanical properties of inhomogeneous structures from the viewpoint of atomic scale to understand the mechanical behaviour of microscopic materials. Thus, the mechanical properties of local regions with inhomogeneous and non-uniform structure have been intensively investigated on the basis of the atomic stress and the elastic constant. In this study, in order to examine the mechanical behaviour of local regions under external load, we conduct atomic tensile simulations on a Sgr5-(013) tilt grain boundary in aluminium using the many-body effective medium theory potential function. The structure of the grain boundary consists of two types of atomic layers, one of which has a higher elastic constant near the grain boundary and the other has a lower one. Although the grain boundary consists of identical atomic species, the local region near the grain boundary displays the mechanical properties similar to those of a composite material.
In this paper, we propose a semi-analytical method to evaluate a lower bound for the elastic properties of bimaterial interfaces such as grain boundaries for example. In this method, interfacial elastic properties are estimated analytically in terms of the inter-atomic potentials of the material. To evaluate the lower bound of these analytical expressions, one only needs to know the relaxed configuration of the atomic assembly near the interface of interest. In comparison with other existing methods, the semi-analytical method developed here significantly reduces the amount of computation. More importantly, because of the analytical nature of the method, it provides a much better tool to study and have a fundamental understanding of the elastic behavior of bimaterial interfaces.
In this paper, a semi-analytic method is developed to compute the surface elastic properties of crystalline materials. Using this method, surface elastic properties, such as the intrinsic surface energy density, intrinsic surface stress and surface elastic stiffness, are given analytically in terms of the inter-atomic potentials of the material. To evaluate these analytical expressions, one needs to know only the equilibrium (or relaxed) positions of the atoms near the free surface. The relaxed positions of the atoms near the free surface can be obtained by conducting a standard molecular simulation of a free surface. In comparison with existing methods, the semi-analytical method developed here reduces the amount of computation significantly. More importantly, because of the analytical nature of the method, it can provide us with a much better understanding of the surface characteristics. To illustrate the methodology, low-index surfaces (111), (100) and (110) of face-centered cubic metals Cu, Ni, Ag, and Pd are considered. A full set of data on surface elastic properties including the intrinsic surface energy, the intrinsic surface stress tensor, and the surface elastic stiffness tensor have been computed and are reported here for these materials. It is believed that such a complete set of surface property data for these materials are new to the literature.
This Chapter presents an overview of atomistic theory of grain boundary diffusion with emphasis on the understanding of diffusion mechanisms. After a brief introduction to the methodology of atomistic simulations, we discuss recent results for the interaction of vacancies and interstitials with grain boundaries and their migration within the grain boundary core. The interaction of point defects with grain boundaries is more complex than it was expected from earlier work. Point defects can alter local grain boundary structure, delocalize in it, or be mechanically unstable. At high temperatures they can strongly interact with each other and further increase the structural disorder. Vacancies and interstitials are equally important in grain boundary diffusion. They can move in the boundary core by single-atom jumps or by collective transitions involving a group of several atoms. Internal stresses and especially their gradients that always exist in grain boundaries are proposed to be responsible for the unusual behavior of point defects and for the multiplicity of their diffusion mechanisms. The calculated diffusion coefficients in Cu and Al grain boundaries reproduce many features observed experimentally, such as diffusion anisotropy, strong structural dependence of diffusion mechanisms, and correlation between grain boundary diffusion and grain boundary energy. The impact of these findings on the understanding of diffusion in nanocrystalline materials is discussed.
We present a method to calculate macroscopic bulk stresses in tetrahedral amorphous carbon (ta-C) films grown by a realistic atomic scale simulation of ion-beam deposition. Similar to real as-deposited films, the simulated films have a high content of sp3 bonded atoms and large intrinsic compressive stresses. Deriving atomic stresses from an interatomic potential and averaging them over slices inside the film, we show that the average stresses in the inner film region converge to realistic values (∼10 GPa) as the thickness of the slices exceeds 1 nm. The analysis of stress variation with depth reveals that in amorphous films deposited with low energy (20–30 eV) ions the highest compressive stress is attained in the region of steady-state growth, while in films grown with 80 eV ions it reaches a maximum in an intermediate layer adjacent to the crystalline substrate. The transition from graphitic carbon to ta-C is found to occur at a threshold stress of about 13 GPa.
Atomistic and electronic structures in the process of intergranular neck formation of nanocrystalline silicon carbide (SiC) have been investigated by a tight-binding molecular dynamics method. An order-N parallel algorithm is employed for efficient calculations of electronic energy and forces. We find that a defect-free neck is formed between SiC nanocrystals aligned along [112] axis at T
Theoretical prediction of effective properties for multiphase material systems is very important not only to analysis and optimization of material performance, but also to new material designs. This review first examines the issues, difficulties and challenges in prediction of material behaviors by summarizing and critiquing the existing major analytical approaches dealing with material property mode ling. The focus then shifts to some recent advances in numerical methodology that are able to predict more accurately and efficiently the effective physical properties of multiphase materials with complex internal microstructures. A random generation-growth algorithm is highlighted for reproducing multiphase microstructures, statistically equivalent to the actual systems, based on the geometrical and morphological information obtained from measurements and experimental estimations. Then a high-efficiency lattice Boltzmann solver for the corresponding governing equations is described which, while assuring energy conservation and the appropriate continuities at numerous interfaces in a complex system, has demonstrated its numerical power in yielding accurate solutions. Various applications are provided to validate the feasibility, effectiveness and robustness of this new methodology by comparing the predictions with existing experimental data from different transport processes, accounting for the effects due to component size, material anisotropy, internal morphology and multiphase interactions. The examples given also suggest even wider potential applicability of this methodology to other problems as long as they are governed by the similar partial differential equation(s). Thus, for given system composition and structure, this numerical methodology is in essence a model built oil sound physics principles with prior validity, without resorting to ad hoc empirical treatment. Therefore, it is useful for design and optimization of new materials, beyond just predicting and analyzing the existing ones. (C) 2008 Elsevier B.V. All rights reserved
We initiate the development of a theory of the elasticity of nanoscale objects based upon new physical concepts which remain properly defined on the nanoscale. This theory provides a powerful way of understanding nanoscale elasticity in terms of local group contributions and gives insight into the breakdown of standard continuum relations. We also give two applications. In the first, we show how to use the theory to derive a new relation between the bending and stretching properties of nanomechanical resonators and to prove that it is much more accurate than the continuum-based relations currently employed in present experimental analyses. In the second, we use the new approach to link features of the underlining electronic structure to the elastic response of a silicon nanoresonator. Comment: 11 pages, 5 figures
This paper focuses on aspects of fracture mechanics of interface cracks in anisotropic bimaterials. In this case there is a coupling of all three crack-tip fracture modes in the natural interface-crack coordinate system, whereas in the isotropic case, mode 3 is decoupled from modes 1 and 2. This paper intends to shed light on how to interpret the crack-tip fields which are given explicitly along the interface in terms of two (real 3 x 3) bimaterial matrices W and D. A matric function Y is defined in terms of W and D which determines the coupling and oscillations in the crack-tip fields. Explicit expressions for the crack-tip fields and the associated SIFs are given as well as for the energy release rate. The finite (Griffith) interface crack is considered in detail.
AN ELASTOPLASTIC solution for the interface crack with contact zones is presented in this paper. The elastic asymptotic solution is discussed first and approximate expressions for the Comninou stress intensity factors of a Griffith interface crack are obtained. It is shown that the elastic asymptotic solutions based on the assumption of a closed crack tip predict material interpenetration : material elements near the crack tip are "turned inside out" and cross the bimaterial interface. The problem of a crack along the interface of an elastoplastic material and a rigid substrate is analyzed in detail. Any contact between the crack surface and the substrate is assumed to be frictionless. An elastoplastic asymptotic solution for a closed crack tip under plane strain conditions is presented. It is shown that the asymptotic solution is separable in r and 0, where (r, 0) are polar coordinates at the crack tip. A boundary layer approach is used to study the near tip elastoplastic fields under small-scale yielding conditions. The finite element method is used for the solution of the small-scale yieldmg problem and the validity of the developed plastic asymptotic solution is verified numerically. Large-scale yielding solutions for a Griffith interface crack with contact zones are also presented.
Grain boundaries and dislocations are the most common extended defects in crystalline materials. The principal difference between them is that grain boundaries are planar imperfections which are not accompanied by long-range strain and stress fields, while dislocations are line defects that produce long-range elastic strains and stresses in the material. They both strongly affect, and to a great extent control, the physical and mechanical properties of materials. This is the reason why both dislocations and grain boundaries, and also more general types of interfaces, have been studied very extensively throughout the development of materials science. A common feature of both grain boundaries and dislocations is that their basic characteristics are of crystallographic, geometrical nature. Indeed, in earlier structural studies these characteristics were the main topic of experimental and theoretical investigations (Bollman, 1970; Hirth and Lothe, 1982; Sutton, 1984; Sutton and Balluffi, 1995). In the case of dislocations they were closely linked with studies of their elastic fields and associated long-range interactions (Nabarro, 1967; Hirth and Lothe, 1982). Since crystallographic attributes of dislocations and grain boundaries have to be defined prior to any study of their structures and properties we summarize these characteristics briefly in the following section.
Angularly-dependent, many-body potentials for the bond order of saturated and unsaturated bonds are derived within Tight Binding Hückel theory. These potentials should prove invaluable for the atomistic simulation of semi-conductors and transition metals over the wide range of co-ordination observed from clusters through to the bulk.
This chapter presents theoretical foundations about elastic behavior of composite materials. A composite material is a heterogeneous solid continuum that bonds together a number of discrete homogeneous continua, each of which has a well-defined sharp boundary. It is understood that the bonding at the interfaces (and the continuity of each region) remains intact in the present circumstances where the entire mixture is to be placed in an equilibrated state of infinitesimal elastic strain by external loads and constraints. Each separate homogeneous region has its characteristic tensor of elastic moduli (in the stress–strain relation), which when anisotropic reflects a particular alignment of the crystallographic axes relative to the fixed Cartesian ones. A single phase consists of all those regions that share the same tensor of elastic moduli and perhaps also the same alignment of a geometrical shape and of crystallographic axes. This chapter presents an analysis of tensors and elastic behavior. The elastic field of a composite body is discussed. The overall elastic behavior of a composite body is also explained in detail.
This chapter focuses on variational and related methods for the overall properties of composites. A wide range of phenomena that are observable macroscopically are governed by partial differential equations that are linear and self-adjoint. This chapter is concerned with such phenomena for materials, such as fiber-reinforced composites or polycrystals, whose properties vary in a complicated fashion from point to point over a small, “microscopic” length scale, while they appear “on average” (that is, relative to the larger, macroscopic scale) to be uniform. This chapter treats the elastic behavior of composites, and emphasizes that a number of other properties (conductivity, viscosity of a suspension, etc.) are described by the same equations. Extensions to viscoelastic and thermoelastic behavior are presented, for both of which the variational characterization given is believed to be new. Problems, such as the resistance to flow of viscous fluid through a fixed bed of particles are mentioned, and a model problem that involves diffusion is presented in some detail. This displays the same difficulty in relation to divergence of an integral and is one problem of this type that has so far been approached variationally. Methods related to the Hashin–Shtrikman variational principle are also described in the chapter.
The main objective of the majority of the microscopic atomic level studies of interfaces has been investigation of the dominant atomic level structural features and their relation to geometrical and crystallography: characteristics of grain boundaries. At the same time extensive studies of mechanical properties of materials with interfaces have been made in the framwork of continuum mechanics but without taking either structure or properties of the interfaces into account. In this paper we attempt, at least partially, to bridge the gap between atomistic and continuum approaches. Since the most fundamental properties of materials entering continuum treatments are the elastic moduli, we demonstrate first that atomic-level elastic properties of interfaces can be rigorously defined and that the fourth-order tensor of effective elastic moduli of grain boundary regions can be determined in atomistic simulations. We then proceed to employ these concepts in the continuum analysis of long wavelength phonons corresponding to acoustic interface waves. It is then shown that only when the local elastic properties of interfaces are incorporated into the contiuum model, via boundary conditions derived directly from the atomic-level elastic properties of interfaces, are the interfacial waves deduced from this model in good agreement with the phonon calculations.
The Symposium on which this volume is based was conceived as a timely expression of some of the fast-paced developments occurring throughout materials science and engineering. It focuses particularly on those involving modern computational methods applied to model and predict the response of materials under a diverse range of physico-chemical conditions. The current easy access of many materials scientists in industry, government laboratories, and academe to high-performance computers has opened many new vistas for predicting the behavior of complex materials under realistic conditions. Some have even argued that modern computational methods in materials science and engineering are literally redefining the bounds of our knowledge from which we predict structure-property relationships, perhaps forever changing the historically descriptive character of the science and much of the engineering.
Stability, in the sense of limits to the uniqueness of solutions to quasi-static boundary value problems, is investigated for two semi-infinite solids bonded along a planar interface. The interface is characterized by a traction-displacement jump relation, so that dimensional considerations introduce a characteristic length. Stability is addressed in terms of the existence of certain stationary waves. Complex variable methods are exploited to obtain an explicit solution when the interface instability precedes bulk localization. Particular cases are analyzed that illustrate a range of behaviors. Under certain conditions, a minimum wavelength for the instability mode is predicted that is at least one to two orders of magnitude larger than the interface constitutive characteristic length. A post-bifurcation analysis, carried out for linear elastic material behavior, shows how the instability plays a role in the transition from a uniform mode of separation to crack-like behavior. The results obtained suggest a mechanism for the size dependence of the failure mode.
The relation between atomic structure and elastic properties of grain boundaries is investigated theoretically from both atomistic and continuum points of view. A heterogeneous continuum model of the boundary is introduced where distinct phases are associated with individual atoms and possess their atomic level elastic moduli determined from the discrete model. The effective elastic moduli for sub-blocks from an infinite bicrystal are then calculated for a relatively small number of atom layers above and below the grain boundary. These effective moduli can be determined exactly for the discrete atomistic model, while estimates from upper and lower bounds are evaluated in the framework of the continuum model. The complete fourth-order elastic modulus tensor is calculated for both the local and the effective properties. Comparison between the discrete atomistic results and those for the continuum model establishes the validity of this model and leads to criteria to assess the stability of a given grain boundary structure. For stable structures the continuum estimates of the effective moduli agree well with the exact effective moduli for the discrete model. Metastable and unstable structures are associated with a significant fraction of atoms (phases) for which the atomic-level moduli lose positive definiteness or even strong ellipticity. In those cases, the agreement between the effective moduli of the discrete and continuum systems breaks down.
The contribution of plastic deformation to the effective work of fracture is computed for a crack lying along one of the interfaces of a thin ductile layer joining two elastic solids. A model is proposed for the joint whose major parameters are the layer thickness, the elastic-plastic properties of material in the layer, and the work of separation and peak separation stress of the local interface fracture process. A symmetric mode I loading of the joint is considered under conditions where the thickness of the layer and the extent of the plastic zone are small compared with the crack length. The crack growth resistance behaviour is computed, with special emphasis on the steady-state work of fracture. The role of the layer thickness in the development of the plasticity contribution to toughness is detailed. Plastic dissipation is fully realized for layers above a certain thickness, characteristic of a plastic zone dimension, and is negligible when the layer is thin relative to this dimension. Other factors which may effect the effective toughness of the joint, such as modulus mismatch of the layer and adherends and residual stress in the layer, are discussed together with limitations and possible extensions of the model.
Using the approach of Finnis and Sinclair, N-body potentials for copper, silver, gold and nickel have been constructed. The total energy is regarded as consisting of a pair-potential part and a many body cohesive part. Both these parts are functions of the atomic separations only and are represented by cubic splines, fitted to various bulk properties. For the noble metals, the pair-potentials were fitted at short range to pressure-volume relationships calculated by Christensen and Heine so that interactions at separations smaller than that of the first-nearest neighbours can be treated in this scheme. Using these potentials, point defects, surfaces (including the surface reconstructions) and grain boundaries have been studied and satisfactory agreement with available experimental data has been found.
If a composite is modeled as a randomly inhomogeneous medium, an ”overall modulus operator” relates the ensemble average of the stress to that of the strain. This paper reviews methods for estimating the overall modulus operator when only a limited amount of statistical information on the composite is available. The most securely based estimates come from bounds on the energy. Such bounds induce bounds on the components of the overall modulus operator and have a precise status, independent of the pointwise quality of the approximating fields from which they are derived. Other approximations are also outlined, however; these include perturbation series, estimates obtained from making ad hoc closure assumptions, and self-consistent estimates. The paper concludes with a discussion of possible future developments.
Composite materials are reviewed fromthe applied mechanics and engineering science point of view. Analyses are presented of composite materials properties such as: elasticity, thermal expansion, mositure swelling, viscoelasticity, conductivity (which includes, by mathematical analogy, dielectrics, magnetics, and diffusion) static strength, and fatigue failure.
This volume contains the Proceedings of the Symposium on Structure/Property Relationships for Metal/Metal Interfaces held at the 1991 Spring Meeting of the Materials Research Society. The papers published here, address this topic by means of experimental and theoretical investigations on a variety of interfacial systems that are found or can be fabricated in metals. The central theme of the Symposium was the relationship between the physical properties of metallic materials containing interfaces and the detailed atomic structure of those interfaces. When categorized by an interest in interfacial structure, these fields include metal/vacuum surface physics and surface layer characterization, x-ray diffractometry of films and superlattices, TEM characterization of orientational relationships in superlattices, grain boundaries, and dissimilar metal interfaces, and analytical microscopy of grain boundaries and dissimilar metal interfaces. As categorized by an interest in properties, these fields include elastic and plastic properties of superlattices, grain boundaries, and generic interfaces, dislocation motion through interfaces, fracture, segregation, and precipitation at grain boundaries and at dissimilar metal interfaces, x-ray optical properties of multilayers, and magnetic properties of superlattices.
The graphitic phases of C and Si are studied with the use of the pseudopotential local-density-functional approach. For graphite, good agreement with experiment is obtained for the in-plane lattice constant, interlayer spacing, isotropic bulk modulus, and graphite-diamond structural energy difference. Graphitic Si has relatively weak bonding and its formation is unlikely since its energy is 0.71 eV/atom higher than the diamond phase and a large negative pressure of -69 kbar is required.
We demonstrate that not only the static structural properties but also the crystal stability and pressure-induced phase transformations in solids can be accurately described employing an ab initio pseudopotential method within the local-density-functional formalism. With the use of atomic numbers of constituent elements and a subset of crystal structures as the only input information, the calculated structural properties of Si and Ge are in excellent agreement with experiment.
Thermodynamics of Crystals is a gold mine of a references bargain with more derivations of useful equations per dollar, or per page, than almost any other book I know. Useful to whom? To the solid state physicist, the solid state chemist working the geophysicist, the rock mechanic, the mineral physicist. Useful for what? For lattice dynamics, crystal potentials, band structure. For elegant, rigorous, and concise derivations of fundamental equations. For comparison of levels of approximation. For some data and physical insights, especially for metals and simple halides. This book is a reissue, with some changes and additions, of a 1970 treatise. It ages well, since the fundamentals do not change.
A simple form of multi-ion interaction has been constructed for the purpose of atomistic simulation of transition metals. The model energy consists of a bonding term, which is the square-root of a site density ρi, summed over atoms i, and a repulsive pairwise term of the form The site density ρi is defined as sum over neighbouring sites j of a cohesive potential (R ij). Both V and are assumed to be short-ranged and are parameterized to fit the lattice constant, cohesive energy and elastic moduli of the seven body-centred-cubic (b.c.c.) transition metals. The result is a simple model which, unlike a pair-potential model, can account for experimental vacancy-formation energies and does not require an externally applied pressure to balance the “Cauchy pressure”.
Theoretical and experimental work on the atomic structure of grain boundaries in metals and ionic crystals is reviewed critically. The dislocation and O-lattice models are described and their usefulness is assessed with reference to a worked example. The models derived from computer simulations of the atomic structures of grain boundaries are described and critically assessed, particularly with regard to their predictive capacities. Key transmission electron microscope observations of lowenergy boundary planes and grain-boundary dislocations are discussed. The difficultles in interpreting high-resolution electron-microscope images of grain -boundary atomIC structures are described and some recent results are discussed in the light of these remarks and computer calculations of grainboundary atomic structures. Finally, an important result from X-ray diffraction studies of grain boundaries is analysed.
The effects of non-central forces in atomistic studies of grain boundaries in molybdenum and tungsten, the transition metals with half-filled d-band, are investigated. For this purpose we have used two different types of potential which include different number of moments of the local density of electronic states when evaluating the total energy: the central-force Finnis-Sinclair potentials which include the scalar second moment and the potentials constructed by Carlsson which include the fourth and the matrix second moments. The energy terms associated with these two moments represent non-central interactions and assure that the bcc-fcc structural energy difference is reproduced with good accuracy. For the three boundaries studied, the non-central forces have been found to be very important in determining the lowest energy structures. In particular, the energy differences between multiple structures depend on specific orientations and geometries of the atomic clusters at and near the interface. On the other hand, central-force potentials favour structures with atomic separations close to those found in the bulk with no regard to bond orientation. As a consequence the lowest-energy structures predicted by the two potential schemes differ in details in both the local atomic relaxations and the magnitude of the rigid-body displacements of the grains, although many general features of the boundary structures remain the same, independent of the potentials used. The calculations also show that it is not possible to identify the major non-central contribution with the fourth moment alone. Thus inclusion of both the matrix second moment and the fourth moment energy contributions is essential for an appropriate description of non-central atomic interactions.
The theoretical transition between basic properties of elastoplastic media at two levels of description is examined rigorously. At the micro-level the material response is heterogeneous, whereas at the macro-level it appears homogeneous. A broad class of constitutive relations is envisaged, and no restriction is placed on the magnitude of deformations and rotations at the micro-level. The investigation is concerned with quadratic differential forms that feature prominently in constitutive analyses, and is complementary to a previous study of bilinear differential forms. A principal objective is to access the transmissibility of measure-invariant inequalities from one level to the other.
The relationship between atomic structure and elastic properties of grain boundaries is investigated from both discrete and continuum points of view. The complete fourth-order tensors of both the atomic-level and the effective elastic moduli are defined for the discrete system, where the latter corresponds to sub-blocks from an infinite bicrystal and are calculated here for a relatively few atomic layers above and below the grain boundary. Then, a heterogeneous continuum model of the boundary is introduced where distinct phases are associated with individual atoms and possess their atomic-level moduli. Only estimates (upper and lower bounds) of the effective moduli can be determined for the continuum model. Comparison between the atomistic results and those for the continuum model establishes the validity of this definition of elastic properties for heterogeneous structures at atomic scales. Furthermore, these comparisons as well as algebraic properties of the fourth-order tensor of moduli lead to criteria to assess the stability of a given grain boundary structure.
A general algebraic framework is developed for characterizing the set of possible effective tensors of composites. A transformation to the polarization-problem simplifies the derivation of the Hashin-Shtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials. A general connection is established between two methods for bounding effective tensors of composites. The first method is based on the variational principles of Hashin and Shtrikman. The second method, due to Tartar, Murat, Lurie, and Cherkaev, uses translation operators or, equivalently, quadratic quasiconvex functions. A correspondence is established between these translation operators and bounding operators on the relevant non-local projection operator, T1. An important class of bounds, namely trace bounds on the effective tensors of two-component media, are given a geometrical interpretation: after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors. A wide class of translation operators that generate these bounds is found. The extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors. These extremal translations generate attainable trace bounds even when the tensors of the two-components are not well ordered. In particular, they generate the bounds of Walpole on the effective bulk modulus. The variational principles of Gibiansky and Cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by Golden and Papanicolaou. An isomorphism is shown to underlie their variational principles. This isomorphism is used to obtain Dirichlet-type variational principles and bounds for the effective tensors of general non-selfadjoint problems. It is anticipated that these variational principles, which stem from the work of Gibiansky and Cherkaev, will have applications in many fields of science.
It is most desirable to understand the structure and chemistry of the internal interfaces for all classes of materials since the materials' properties often depend on the properties of the interfaces which, in turn, are controlled by their structure and chemistry. In contrast to surface science, there exist only a few techniques for studying the structure and chemistry of internal interfaces. One of the most powerful techniques seems to be transmission electron microscopy (TEM) by which short segments of interfaces can be analyzed. In high-resolution electron microscopy (HREM) a direct image is formed of the projection of the interfaces. A simple analysis of HREM micrographs is not possible owing to the complex image forming processes within HREM. In addition to experimental investigations, calculations of the structures must be performed using material specific interatomic potentials. From the calculated structure, HREM images must be simulated for the specific imaging conditions. The experimental micrographs must be compared to simulated images. An agreement between experimental micrographs and the simulated images results in the best possible atomistic configuration. A quantitative measure for this agreement is the difference image, D, between the experimental micrograph and the simulated image. Best agreement is reached if only the noise is visible in the difference image D. Analytical electron microscopy with high-spatial resolution (typical probe size -Al2O3. It should be emphasized, however, that the TEM techniques could also be applied to internal interfaces in different boundaries.
In recent years it has become well established that fast diffusion along grain boundaries plays a key role in many important
metallurgical processes including cases where net mass is transported along boundaries which act as sources and/or sinks for
the fluxes of atoms. In addition, considerable advances have been made in understanding grain boundary structure, and new
techniques have become available for studying kinetic phenomena in grain boundaries. This lecture will attempt to review our
current knowledge of the atomistic mechanisms responsible for these grain boundary diffusion phenomena. Relevant aspects of
the structure of grain boundaries and the point and line defects which may exist in grain boundaries are described first.
The important experimental observations are then discussed. Diffusion models are then taken up, and it is concluded that the
atomic migration occurs by a point defect exchange mechanism which, in at least the vast majority of boundaries in simple
metals, most likely involves grain boundary vacancies. The grain boundary sources and/or sinks required to support divergences
in the atomic (vacancy) fluxes are grain boundary dislocations. Phenomena therefore occur which resemble the Kirkendall Effect
in the bulk lattice in certain respects. Additional topics are discussed which include effects of boundary structure on boundary
diffusion and the question of whether or not boundary diffusion is faster along migrating than stationary boundaries.
A simple empirical tight-binding model for silicon is propounded and used to compute the atomic and electronic structures of three symmetrical tilt grain boundaries and the intrinsic stacking fault. The ability of the model to describe silicon in a variety of crystal structures is tested and it is shown to be satisfactory for simulating defects in the diamond structure. The effect of charge transfer on the energy and stability of the grain boundaries is assessed. Interatomic forces and energies are computed in real space using a rotationally invariant formulation of the recursion method. Five proposed reconstructions of the (112̄) symmetrical tilt boundary are studied in detail and good agreement is achieved with results from electron microscopy and diffraction. The (13̄0) and (111) symmetrical tilt boundaries have also been modelled successfully. Comparison is made between the computed electronic structures of the boundaries, reported in this work and by other authors, and experimental measurements of the densities of states at grain boundaries. The existence of band tails and midgap continua in the experimental measurements and the absence of both of these features in the models are two notable points of disagreement. Some fundamental questions about localisation of electronic states at grain boundaries are raised.
This paper describes a finite element method for simulating migration of interfaces (e.g. phase and grain boundaries) in materials. The method is built on a classical theory. Each individual grain is in an equilibrium state; interface tension and bulk phase chemical potential constitute the free energy. An interface migrates—as atoms break from one grain, cross the interface, and attach to the other grain—at a velocity proportional to the free energy reduction per unit volume of atoms crossing the interface. We express this theory in a weak statement, model the interfaces with finite elements, and update nodal positions incrementally. The variations of the free energy, associated with the virtual motions of the nodes, define the generalized forces. The weak statement connects the generalized forces and the nodal velocities with a viscosity matrix. The method takes into account large interface shape changes, interface tension anisotropy, and non-equilibrium triple junctions (if present). We illustrate the method with examples including grooving on a polycrystal surface, grain growth in a thin film, and facet formation of a single crystal particle.
An International Symposium on the Structure of Chemistry of Grain Boundaries was held in Bechyně, Czechoslovakia, on May 6–10, 1991, which brought together a small but very active group of leading scientists from Europe, USA and Japan. In this report we summarize the main topics discussed at the symposium and highlight in particular those areas which are likely to play a leading role in future research on grain boundaries (GBs) and other interfaces. The first part of the report deals with theoretical investigations of the structure and properties of GBs and emphasizes new approaches related to the development of ab intion methods and new thermodynamic approaches. The second part is devoted to observations of GB structures, with the main emphasis on electron microscopy, and the third section deals with the chemistry of GBs, pricipally with segregation phenomena and GB phases.
The structure and composition of metal/ceramic interfaces play an important rôle for the properties of composites, for the bonding of bulk metals (or metallic alloys) to bulk ceramics, in electronic packaging, and for the properties of oxide scales on metals (or metallic alloys) formed after high-temperature corrosion. In this paper the possibilities of high-resolution transmission electron microscopy (HRTEM) and analytical electron microscopy (AEM) will be summarized and recent developments discussed. Those advancements encompass quantification of HRTEM data by image processing and studies of specific components of the interfaces by investigations of the near-edge fine structures (ELNES) of energy loss spectra. The techniques will be applied to the , and interfaces, respectively.
Empirical interatomic potentials permit the calculation of structural properties and energetics of complex systems. A new approach for constructing such potentials, by explicitly incorporating the dependence of bond order on local environment, permits an improved description of covalent materials. In particular, a new potential for silicon is presented, along with results of extensive tests which suggest that this potential provides a rather realistic description of silicon. The limitations of the potential are discussed in detail.
An alternative parametrization is given for a previous empirical interatomic potential for silicon. The new potential is designed to more accurately reproduce the elastic properties of silicon, which were poorly described in the earlier potential. The properties of liquid Si are also improved, but energies of surfaces are less accurate. Detailed tests of the new potential are described.
Phonon spectra of bicrystals with relaxed grain-boundary structure display a variety of localized modes including long-wavelength acoustic modes. Continuum solutions for localized waves that incorporate atomic-level elastic properties of the interface via discontinuity relations agree well with the latter modes. In contrast, classical solutions that depend only on bulk elastic properties do not. This demonstrates that the distinct atomic structure of the interface is a controlling factor, and it is shown how local, atomic-level properties can be incorporated into continuum analyses of interfacial phenomena.