No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Energy of grain boundaries between cusp misorientations
Abstract
The disclination model of high angle grain boundaries proposed before is used to understand the energy-angle relations between cusp misorientations. It is found that the model predicts the correct relationship within experimental error with only one adjustable parameter which resembles the Burgers vector of dislocations at low angles. Experimental data for [100] symmetric tilt boundaries in Cu and Al and for [110] symmetric tilt boundaries in Al are used for illustration.
... The stresses associated with an edge dislocation are represented using the classical continuum formulation, and low-angle symmetric tilt grain boundaries can be represented using repeating edge dislocations [17]. For the present work, disclination dipoles are used to construct tilt boundaries (via the Disclination Structural Unit Model or DSUM [18][19][20]) for a wide range of tilt angles. This avoids the limitation of the dislocation model, which is limited to low-angle boundaries. ...
... H is the period of the grain boundary and k ¼ pd 0 n H , where d 0 n is the distorted length of the minority structural unit. f ðkÞ is calculated as [18,19]: ...
... Tilt grain boundaries Intrinsic stress fields for general tilt boundaries are constructed via a series of disclination dipoles, based on the disclination structural unit model (DSUM)[18,26,27]. The resulting stress field equations are given by[20 ...
... The MD simulations capture the atomistic scale deformation signatures that govern the failure behaviors of the graphene GBs, which in turn help understand the origin of the variations of failure strength against loading angle shown in Fig. 8. We find that the graphene fracture initiates from bond breaking at the disclination dipole [69], followed by the coalescence of such defects due to bond breaking into a long crack, whose further propagation leads to the failure of the graphene. In other words, the failure strength of the GB is governed by the bond breaking at the disclination dipole. ...
... Consider an AC GB made of an array of disclination dipoles [69], as illustrated in Fig. 11a. For the ith disclination dipole with the center of its pentagon at A(x i A , y i A ) and the center of its heptagon at B(x i B , y i B ), the induced stress field in the graphene at (x; y) is given by: ...
Understanding the tensile strength of graphene grain boundaries (GBs) is crucial for correlating the mechanical properties of two dimensional polycrystalline graphene with its atomic defect structure, a key to the success of large area graphene in many promising applications. Existing modeling studies mainly focus on the deformation and fracture of graphene GBs under tension that is perpendicular to the GBs. In reality, however, when a polycrystalline graphene is subject to a simple tension, random distribution of GBs in the graphene leads to arbitrary in-plane loading conditions of the GBs that cannot be fully understood with existing knowledge. To this end, we carry out systematic molecular dynamics (MD) simulations and also delineate a continuum mechanics model to investigate the failure strength of graphene GBs under tension in all possible loading directions. Particular focus is placed on quantitatively deciphering the interplay between GB misorientation angle and loading angle, and their effects on the failure strength of graphene GBs. Prediction from the continuum mechanics model based on a disclination dipole theory agrees well with the results from MD simulations. In this sense, the present study offers important insights on a better understanding of the mechanical properties of large area polycrystalline graphene.
... Disclinations were originally conceived as rotational lattice defects to complement dislocations in the classic work of Volterra [61] and have been used as a component in the description of the structure of grain boundary interfaces in crystalline materials [62,63]. Disclinations contribute to lack of closure over a Burgers circuit manifested by gradients of lattice curvature. ...
... Recall from Section 1.2 that the undeformed structure of high-angle grain boundary interfaces may be described using disclination dipoles along with the structural unit model [62][63][64][65][66][67][68]. In this framework, the minority structural units are represented as wedge disclination dipoles while the strength of each wedge disclination is defined by the change in the angle between neighboring dissimilar structural units. ...
... In crystalline media, grain boundaries are rotational defects resulting in certain discontinuities of the elastic/plastic strain and/or curvature fields. Modeling efforts at describing grain boundaries include dislocation-based and disclination-based approaches (see respectively (Frank, 1950;Bilby, 1955) and (Li, 1972;Shih and Li, 1975;Gertsman et al., 1989)) and atomistic simulations (Sutton et al., 1983). In dislocation-based models, surface-dislocation densities are considered as the source of disorientation between grains, i.e. an arbitrary disorientation is accommodated by an appropriate distribution of surface-dislocations. ...
... Being themselves rotational defects, disclinations may seem to be more appropriate than dislocations for grain boundary modeling (Li, 1972). Discrete disclination dipole walls were indeed introduced to represent grain boundaries (Li, 1972;Shih and Li, 1975;Gertsman et al., 1989). These models are built on linear arrays of discrete disclination dipoles, for which closed-form expressions of the elastic energy, strain and curvature fields were first derived by Huang andMura (1970) andde Wit (1973). ...
The continuity vs discontinuity of the elastic/plastic curvature & curvature rate, and strain & strain rate tensors is examined at non-moving surfaces of discontinuity, in the context of a field theory of crystal defects (dislocations and disclinations). Tangential continuity of these tensors derives from the conservation of the Burgers and Frank vectors over patches bridging the interface, in the limit where such patches contract onto the interface. However, normal discontinuity of these tensors remains allowed, and Kirchhoff-like compatibility conditions on their normal discontinuities across the concurring interfaces are derived at multiple junctions. In a simple plane case and in the absence of surface-disclinations, the compatibility of the normal discontinuities in the elastic curvatures assumes the form of a Young’s law between the grain-to-grain disorientations and the sines of the dihedral angles. Complete continuity of the plastic strain rate tensor at triple junctions also derives from the compatibility of the normal discontinuities in the plastic strain rates in such conditions.
... Later on, the disclination model was modified in Ref. [98] to find the energies of tilt GBs between so-called cusp misorientations, for which GBs possess local minima of energy because of their preferable atomic structure. The next step in applying disclination approach to the analysis of GB properties was accounting for structural units -the elements of GBs of finite length [99][100][101][102]. ...
... The numerical implementation of the slip transmission algorithm is introduced in the Methodology section. To impose the stress field of each GB, the disclination structural unit model (DSUM) [71][72][73] and a model for the incorporation of EGBDs within the DSUM structure [16,74] are coupled with DDD. DSUMbased construction of equilibrium STGBs in terms of wedge disclination dipoles [64,69,75,76] (WDDs) is detailed in the Framework for Modeling Equilibrium and Non-equilibrium Grain Boundaries section. ...
... Generally, disclinations are liable for complementing dislocations in the description of the lattice structure when single-valued elastic rotation fields do not exist, as in polycrystals. Disclination dipoles were shown to provide a good description of grain boundaries [9][10][11][12]. ...
We review the mechanical theory of dislocation and disclination density fields and its application to grain boundary modeling. The theory accounts for the incompatibility of the elastic strain and curvature tensors due to the presence of dislocations and disclinations. The free energy density is assumed to be quadratic in elastic strain and curvature and has nonlocal character. The balance of loads in the body is described by higher-order equations using the work-conjugates of the strain and curvature tensors, i.e., the stress and couple-stress tensors. Conservation statements for the translational and rotational discontinuities provide a dynamic framework for dislocation and disclination motion in terms of transport relationships. Plasticity of the body is therefore viewed as being mediated by both dislocation and disclination motion. The driving forces for these motions are identified from the mechanical dissipation, which provides guidelines for the admissible constitutive relations. On this basis, the theory is expressed as a set of partial differential equations where the unknowns are the material displacement and the dislocation and disclination density fields. The theory is applied in cases where rotational defects matter in the structure and deformation of the body, such as grain boundaries in polycrystals and grain boundary-mediated plasticity. Characteristic examples are provided for the grain boundary structure in terms of periodic arrays of disclination dipoles and for grain boundary migration under applied shear.
... Pentagon-heptagon pairs are analogous to dislocations in bulk crystalline materials and are the most important defects in a 2D hexagonal lattice. From a geometrical perspective, a pentagon-heptagon pair resembles a disclination dipole [129][130][131][132], which consists of two disclinations of opposite signs. In view of this, Wei et al. [26] constructed the stress field of a pentagon-heptagon pair by using the disclination dipole model. ...
The super-high strength of single-layer graphene has attracted great interest. In practice, defects resulting from thermodynamics or introduced by fabrication, naturally or artificially, play a pivotal role in the mechanical behaviors of graphene. More importantly, high strength is just one aspect of the magnificent mechanical properties of graphene: its atomic-thin geometry not only leads to ultra-low bending rigidity, but also brings in many other unique properties of graphene in terms of mechanics in contrast to other carbon allotropes, including fullerenes and carbon nanotubes. The out-of-plane deformation is of a ‘soft’ nature, which gives rise to rich morphology and is crucial for morphology control. In this review article, we aim to summarize current theoretical advances in describing the mechanics of defects in graphene and the theory to capture the out-of-plane deformation. The structure–mechanical property relationship in graphene, in terms of its elasticity, strength, bending and wrinkling, with or without the influence of imperfections, is presented.
... Adaptive Intermolecular Reactive Empirical Bond Order (AIREBO) potential was used in their simulations, and the fracture behavior of graphene was based on the cutoff function value and range of covalent interactions. Wei et al. [69] focused on the fundamental understanding of how defects and defect configurations of disclination dipoles [73][74][75] (resembling pentagon heptagon pairs) formed by 5-7 defects interact in polycrystalline graphene [76] using Molecular Dynamics (MD) simulations based on AIREBO potential. [66,77] The effect of SWD on hydrogenated graphene using AIREBO potential was studied by Verma and Parashar. ...
In this paper, nanoscale mechanical properties and failure behavior of graphene with Stone-Wales defect concentration were investigated using molecular dynamics simulations with the latest ReaxFF C-2013 potential that can accurately capture bond breakages of graphitic compounds. The choice of interatomic potential plays an essential role in capturing the deformation mechanism accurately. Stable configuration of two-dimensional graphene experiences out-of-plane deformation leading to ripples and wrinkles in graphene. It is observed that the mechanical properties such as Young's modulus, ultimate tensile strength, and the fracture strain are dependent on the out-of-plane deformation, temperature, defect concentration, defect orientation, defect layout and loading configuration. It is observed that the post transient phase non-homogenous ripples and wrinkles influence the mechanical properties at low and high defect concentrations, respectively.
... For example, atomic defects such as vacancies, substitutions, or atoms at interstitial positions are not associated to a holonomy, and therefore are not considered topological. For grain boundaries separating regions of different lattice orientations, it has been suggested that they can be described as arrays of dislocations [35][36][37] or disclinations [38][39][40][41][42][43]. ...
We establish a link between the ground-state topology and the topology of the lattice via the presence of anomalous states at disclinations -- topological lattice defects that violate a rotation symmetry only locally. We first show the existence of anomalous disclination states, such as Majorana zero-modes or helical electronic states, in second-order topological phases by means of Volterra processes. Using the framework of topological crystals to construct $d$-dimensional crystalline topological phases with rotation and translation symmetry, we then identify all contributions to $(d-2)$-dimensional anomalous disclination states from weak and first-order topological phases. We perform this procedure for all Cartan symmetry classes of topological insulators and superconductors in two and three dimensions and determine whether the correspondence between bulk topology, boundary signatures, and disclination anomaly is unique.
... Hence, the propagation of plastic slip across adjacent grains is possible at low strain only by the indirect transmission of dislocations in some particular GB facets ( Priester, 2013 ) or with the activation of dislocation sources ( Lee et al., 1990;Malyar et al., 2017 ). A large number of dislocations are then accumulated at or very close to the GBs during the first stages of plastic deformation ( Amouyal et al., 2005;Shih and Li, 1975 ). ...
... Dans le cas des jdg d'angle faible, l'énergie propre de chaque jdg est fonction de sa désorientation [Shih, 1975]. L'énergie des jdg de faible désorientation peut être calculée grâce à la formule de Read-Shockley [Hasson, 1971;Read, 1950;Skipmore, 2004]. ...
REALISATION DE SERIE D4IMAGE EN UTILISANT ecstm FAIRE une etude comparative et definire le role de la microstructure
... GBs are represented in this work using the disclination structural unit model (DSUM) to achieve misorientations θ 12 , θ 23 , and θ 31 [15,[44][45][46]. Disclination dipoles are located periodically along each GB, and the overall misorientation is determined by the strength of the disclinations, the arm length in each dipole, and the spacing among dipoles. ...
Abstract By using a generalized, spatially resolved rate theory, we systematically studied the irradiation-induced diffusion and segregation of point defects near triple junctions. Our model captured not only the formation, growth, and recombination of point defects but also the interaction of these defects with pre-existing defects. We coupled the stress field of the triple junction with defect diffusion via a modified chemical potential. The residual stress fields of grain boundaries and triple junctions are modeled via disclination mechanics theory. By assessing the behavior of 144 triple junctions with vacancy and interstitial defects, we correlated defect-sink efficiencies with key characteristics of triple junctions. For vacancies, the geometric configuration of triple junctions dominated sink efficiency, suggesting that equiaxed grains would resist the accumulation of vacancies more than elongated grains. For interstitials, the sink density of the grain boundaries composing the triple junctions dominated sink efficiency. Hence, the interstitial concentration may be managed by adjusting the structure of the grain boundaries. Overall, we illustrated the complex coupling between pre-existing defects and radiation-induced defects through interaction of their stress fields. This theoretical framework provides an efficient tool to rapidly assess defect management in microstructures.
... The GB energies of the structures for 17 110 STGBs as a function of the GB angles for 27 different at.%C ranging from 0.0058% to 0.1569% (which is more than the solubility of carbon in α-Fe, but is used to ensure the effect of the high carbon concentration) is shown in Fig. 8. The energy of boundary in the simulation is related to parameters such as the relative separation of the two grains and the porosity of the GB [68]. However, it could be studied by continuum approach that the GB is described by structural defects such as dislocations [69]. ...
In this paper, molecular dynamics simulations were used to investigate the effect of the presence of carbon atoms, either in dispersed form or C-rich region, in low-carbon α-Fe containing symmetric tilt grain boundary (STGB) with a boundary plane rotated about the 110 misorientation axis on the number of SIAs and vacancies produced by PKA energies of 3, 5, 7 and 9 keV at 300 K. Results were compared with the SIAs and vacancies produced in pure α-Fe. It was also shown that the presence of GBs in this Fe-C alloy has no effect on the time at which point defects reach to their maximum values at the thermal spike stage. On the other hand, the GBs decrease the number of point defects in comparison to Fe-C without GB planes. It was also concluded that the carbon, either in dispersed form or C-rich region, has no meaningful effect on the number of survived point defects. Furthermore, the number of SIAs is less than the number of vacancies, except at θ=90°. This result was attributed to GB energy because by calculating GB energies as a function of the GB angles for 27 different at.%C ranging from 0.0058% to 0.1569%, a deep cusp was obtained at θ=90°. A sharp rise and fall were observed for the number of SIAs and vacancies at θ=90° misorientation, respectively. The sharp rising becomes smooth for the number of SIAs with increasing EPKA and the sharp falling becomes deeper for the number of vacancies with increasing EPKA.
... The round parentheses in Eq. (1) group self energies of DDWs, energies of nearestneighbor, second-neighbor interactions etc. The self energy of a DDW was calculated by Shih and Li [53], while more general formulae for interaction energies of DDWs displaced with respect to each other along the axes x and y were obtained by Nazarov et al. [54,55] (see, for example Eqs. (2) and (3) in Ref. [55]). ...
Atomic structure of columnar nickel nanocrystals with [1 1 2] column axis having nonequilibrium grain boundaries (GBs) containing extrinsic grain boundary dislocations (EGBDs) and its evolution under oscillating stresses are studied by molecular dynamics method. Energy of GBs as a function of the degree of nonequilibrium is evaluated. It is found that under loading by symmetrically oscillating stresses the nonequilibrium GBs generate lattice dislocations, which travel across the grains and are absorbed by opposite GBs thus resulting in a relaxation of the structure, long-range stress fields and the energy of GBs.
... Comme le démontrent les deux premier types de modèles présenté ci-dessus, la difficulté majeure dans la modélisation de l'énergie des joints de grain provient du calcul de l'énergie du coeur des joints de grains qui peut être désormais calculée à l'aide de simulations numériques à l'échelle atomique. (Li 1972;Shih et al. 1975). Cette dernière est donnée par: ...
Ce travail de thèse est dédie à l'étude de l'effet de taille dans le comportement élastovisocplatsique des matériaux nanocristallin purs à structure cubique face centrée. Ce dernier ce révèle notamment par le non respect de la loi de Hall de Petch se produisant dans cas de matériau à très faible taille de grain. Dans un premier temps, un état de l'art détaille présente le type de matériau considéré, leur structure, procédé de fabrication et comportement. Les effets des mécanismes de diffusion (fluage de Coble. Glissement de Lifshitz) son quantifier par l'introduction d'un modèle composite biphasé. Dans un second temps. L'effet de l'émission de dislocations par les joints de grains est étudie. Une nouvelle loi de comportement est introduite, traduisant de l'effet du mécanisme combinée d'émission et de pénétration de dislocations sur le comportement élasto-viscoplastique des joints de grains. Le modèle se base sur le formalisme des mécanismes thermiquement active et une méthode d'obtention des paramètres basée sur la dynamique moléculaire est introduite. Le comportement macroscopique du matériau est obtenu suite à l'application d'un schéma sécant autocohérent. Ce second modèle est complète par une analyse par éléments finis
... Originally proposed by Shih and Li [34,35] and later improved by Gertsman et al. [36], the disclination structural unit model (DSUM) constructs a (non-favored) grain boundary with a misorientation angle q by decomposing it into a contiguous and alternating sequence of special (favored) m majority and n minority structural units with associated misorientation angles q m and q n respectively such that q m < q < q n . Favored boundaries are grain boundaries that have a structure characterized by a repeating sequence of only one type of structural units. ...
Solute segregation to grain boundaries is considered by modeling solute atoms as misfitting inclusions within a disclination structural unit model describing the grain boundary structure and its intrinsic stress field. The solute distribution around grain boundaries is described through Fermi-Dirac statistics of site occupancy. The susceptibility of hydrogen segregation to symmetric tilt grain boundaries is discussed in terms of the misorientation angle, the defect type characteristics at the grain boundary, temperature, and the prescribed bulk hydrogen fraction of occupied sites. Through this formalism, it is found that hydrogen trapping on grain boundaries clearly correlates with the grain boundary structure (i.e. type of structural unit composing the grain boundary), and the associated grain boundary misorientation. Specifically, for symmetric tilt grain boundaries about the [0 0 1] axis, grain boundaries composed of both B and C structural units show a lower segregation susceptibility than other grain boundaries. A direct correlation between the segregation susceptibility and the intrinsic net defect density is provided through the Frank-Bilby formalism. Overall, the present formulation could prove to be a simple and useful model to identify classes of grain boundaries relevant to grain boundary engineering. © 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
... In contrast to other cases, GBs with θ = 16.4 • (Supplementary Fig. S1d) and 17.9 • (Supplementary Fig. S1e) do not have the same periodic characteristics of pentagon-heptagon defects in the GBs. From a geometrical perspective, a pentagon-heptagon pair resembles a disclination dipole 21,[24][25][26] , which consists of two disclinations of opposite signs. These armchair tilt GBs under present consideration are either composed of an array of uniformly distributed disclination dipoles (Fig. 2a) or an array of disclination dipole clusters (Fig. 2c,e). ...
... C'est cette augmentation d'énergie qui définie la tension interfaciale. Il est admis que cette tension est fonction de la désorientation du joint de grains [19][20][21][22]. ...
This work is dedicated to the study of Grain Boundary Engineering in Ni-based superalloys for aircraft turbine disk. The grain boundary engineering has proven its efficiency for the improvement of the fatigue and creep properties of copper and nickel alloys. This improvement is achieved by performing a succession of room temperature deformations interspaced by heat treatments to modify the distribution of the grain boundary network. The aim of the study is to adapt these processes to high temperature forging of superalloys. An experimental study is achieved to highlight the influence of forging parameters on the grain boundary network. The characterization of the grain boundary network is made through the completion of crystallographic orientation mapping by EBSD. An innovative characterization method based on the discretization of orientation maps in a triple junction network is proposed. This method allows to analyze the connectivity in the grain boundary network with the objective of a correlation with mechanical properties.
... where E c is the energy per unit length due to the dislocation cores, and E s is the surface energy per unit length, and θ al. [151]. Other calculations of GB energy are given by Shih et al. [175] and Shenderova et al. [176]. ...
The emerging field of nanomechanics is providing a new focus in the study of the mechanics of materials, particularly in simulating fundamental atomic mechanisms involved in the initiation and evolution of damage. Simulating fundamental material processes using first principles in physics strongly motivates the formulation of computational multiscale methods to link macroscopic failure to the underlying atomic processes from which all material behavior originates.
A combined concurrent and sequential multiscale methodology is developed to analyze fracture mechanisms across length scales. Unique characterizations of grain boundary fracture mechanisms in an aluminum material system are performed at the atomic level using molecular dynamics simulation and are mapped into cohesive zone models for continuum modeling within a finite element framework. Fracture along grain boundaries typically exhibit a dependence of crack tip processes (i.e. void nucleation in brittle cleavage or dislocation emission in ductile blunting) on the direction of propagation due to slip plane orientation in adjacent grains. A new method of concurrently coupling molecular dynamics and finite element analysis frameworks is formulated to minimize the overall computational requirements in simulating atomistically large material regions. A sequential multiscale approach is advanced to model microscale polycrystal domains in which atomistically-based cohesive zone parameters are incorporated into special directional decohesion finite elements that automatically apply appropriate ductile or brittle cohesive properties depending on the direction of crack propagation. The developed multiscale analysis methodology is illustrated through a parametric study of grain boundary fracture in three-dimensional aluminum microstructures.
... At a finer (discrete) scale, different GB modeling approaches exist in the literature that differ in their ability to describe the initial structure, defect content and energy. These include: (i) dislocation and disclination models [22][23][24][25][26][27]; (ii) geometric theories, such as the coincidence site lattice (CSL), Frank-Bilby and O-lattice theory [28][29][30][31]; and (iii) atomistic simulations, which led to the identification of structural units and their relation with dislocation and disclination models [32][33][34][35]. Experiments can be used to validate these models, such as (a) conventional (bright and dark field) and high-resolution transmission electron microscopy (HRTEM) for the GB structure at small length scales [36][37][38][39][40][41], and (b) measurements of dihedral angles of surface grooves related to the GB energy [42][43][44][45][46][47]. ...
Abstract Grain boundaries play an important role in the mechanical and physical properties of polycrystalline metals. While continuum macroscale simulations are appropriate for modeling grain boundaries in coarse-grained materials, only atomistic simulations provide access to the details of the grain boundary (GB) structure and energy. Hence, a multiscale description is required to capture these GB details. The objective of this paper is to consolidate various approaches for characterizing grain boundaries in an effort to develop a multiscale model of the initial GB structure and energy. The technical approach is detailed using various 〈 1 0 0 〉 , 〈 1 1 0 〉 and 〈 1 1 1 〉 symmetric tilt grain boundaries in copper and aluminum. Characteristic features are: (i) GB energies obtained from atomistic simulations and boundary period vectors from crystallography, (ii) structural unit and dislocation descriptions of the GB structure and (iii) the Frank–Bilby equation to determine the dislocation content. The proposed approach defines an intrinsic net defect density scalar that is used to accurately compute the GB energy for these GB systems. The significance of the present work is that the developed atomistic-to-continuum approach is suitable for realistically inserting the initial GB structure and energy into continuum level frameworks.
... Well known examples of these new mechanisms are the shear coupled boundary migration [Cahn,2006, Gorkaya,2009, Mompiou,2009], or the emission and glide of dislocations from grain boundaries [Tschopp,2008, Tschopp,2008, Van Swygenhoven,2006, Van Swygenhoven,2008]. Modeling efforts for the description of GBs include atomistic simulations [Sutton,1983, Cahn,2006, Tschopp,2008, Tschopp,2008, Van Swygenhoven,2006, Van Swygenhoven,2008], dislocation/disclination-based models [Frank,1950, Bilby,1955, Li,1972, Shih,1975, Yu,1989, Cahn,2006] and mechanical approaches [Wei,2004, Warner,2006, Gurtin,2008, Wei,2009, Zbib,2011]. As shown from atomistic simulations, the grain boundary behavior is core and structure sensitive. ...
A continuum mechanics model is introduced for a core and structure sensitive modeling of grain boundary mediated plasticity. It accounts for long range elastic strain and curvature incompatibilities due to the presence of dislocation and
disclination densities. The coupled spatio-temporal evolution of the crystal defects is also accounted for
by transport equations. Based on atomistic structures, copper tilt boundaries are modelled with periodic sequences of wedge disclination dipoles. Their self-relaxation by transport lead to grain boundary configurations with lower elastic energies, which are compared to molecular statics values. The characteristic internal length inherent to strain gradient elasticity, which relates the weight of couple-stresses to that stresses, is chosen to retrieve the elastic energy obtained by atomistic simulations. This length is shown to be lower than interatomic distances, suggesting that couple-stress elasticity may be appropriate at very low resolution scales only.
... where k ¼ pd B =H for GBs with 0°< h < 21.8°and k ¼ pd C =H for GBs with 21.8°< h < 60°. Functional f is taken as [34]: ...
... However, for the GB energy considered here, the second term in the numerator will always lead to a non-zero rotation of G 1 . On the other hand, if the energy is symmetric about h ¼ 45 (as is the case with the energy given in Fig. 6 of Shih and Li (1975)), the rotation of G 1 will vanish and the grain will shrink purely by migration of C 1 and C 2 . This phenomenon of rotation getting locked has been observed in the MD (Trautt and Mishin, 2014) and phase field simulations (Wu and Voorhees, 2012) when C 1 and C 2 are symmetrically equivalent. ...
... We apply the disclination model to analyze the initial stress induced by hexagon-heptagon rings. The disclination model (Eshelby, 1966;Kleman and Friedel, 2008;Li, 1972;Muskhelishvili, 1953;Shih and Li, 1975;Romanov and Kolesnikova, 2009) has been successfully applied to explain the strength-tilt angle relation for tilted GBs in graphene, as well as the boundary structures in silicon, germanium (Mullner and Pirouz, 1997), biomaterials (Yu and Sanday, 1991) and fullerenes (Kolesnikova and Romanov, 1998). Following Li (1972) and Wei et al. (2012), we consider a pentagon ring in the hexagonal graphene as a positive wedge disclination, and a heptagonal ring as a negative disclination. ...
In two-dimensional polycrystalline graphene, two angular degrees of freedom (DOF) are needed to define a general grain boundary (GB): the misorientation of two grains and the rotation of the boundary line. Via both molecular dynamics simulations and theoretical analysis, we see that the density of GB defects strongly depends on grain misorientation but is insensitive to GB rotation. And reveal the dependence of mechanical properties on grain misorientation and GB rotation in polycrystalline graphene. We find that the dependence of GB normal strength on grain misorientation and GB rotation in graphene stems from the superposition of the stress field induced by a pentagon–heptagon pair itself to that from the interaction between the other defects and the one under consideration. Based on MD simulations and ab initio calculations, we show that failure starts from the bond shared by hexagon–heptagon rings. We then apply continuum mechanics to explain the dependence of GB normal strength on the two angular DOF in graphene with pentagon–heptagon rings. The investigation showed here supplies valuable guidance to develop multiscale and multiphysics models for graphene.
... In the terminology of the differential geometry of continua as well as in the language of engineering, grain boundaries (GBs) in crystalline media are rotational defects resulting in discontinuities of the elastic/plastic strain and/or curvature fields along interfaces. Modeling efforts for the description of GBs include atomistic simulations (Sutton and Vitek, 1983;Cahn et al., 2006), dislocation/disclination-based models (Frank, 1950;Bilby, 1955;Li, 1972;Shih and Li, 1975;Gertsman et al., 1989;Cahn et al., 2006) and mechanical approaches (Wei and Anand, 2004;Warner et al., 2006;Gurtin and Anand, 2008;Wei et al., 2009;Zbib et al., 2011). From the atomistic standpoint, the continuum metrics needed for estimating the strain and curvature tensor fields is still to be built from the positions of the atoms in the interface region, and therefore a description of rotational incompatibility at GBs in continuum terminology is not derived yet. ...
... Of particular interest to the present work are the contributions related to grain and twin boundaries as well as triple junctions (TJ). In early work by Li (1972) and Shih and Li (1975), it was shown that the excess energy of a symmetric tilt boundary can be computed by representing the grain boundary as an alternating sequence of special boundaries separated by disclination dipoles of alternating strengths. Atomistic simulations have shown the connection between the strength and position of disclination dipoles and the atomistic structure of grain boundaries. ...
A disclination-based framework is used to quantify the effect of rotational incompatibility on internal stresses and excess energies in crystalline media in the presence of symmetric tilt boundaries and triple junctions. Also, a new theoretical model for triple junctions, based on the balance of rotational incompatibility at surfaces of discontinuity is introduced. The systems internal energies are obtained first by considering solely the Cauchy stress and elastic strain relationship and then by considering a more general Cosserat-type elastic response, involving couple-stresses and elastic curvature. Comparison between the two models in face centered cubic systems yields quantification of the contribution of rotational defects to internal energy. The work reveals that the curvature and its work conjugate provide for a significant part of the elastic strain energy of symmetric tilt boundaries. In the case of triple junctions, due to screening, such contribution is found to fluctuate significantly. The model is used to exhibit the evolution of the energy of triple junctions built solely from symmetric tilt boundaries as a function of their degrees of freedom. It reveals significant departure from Herrings relationship.
Les effets locaux de deux inhibiteurs organiques de corrosion, le 2-mercaptobenzothiazole (MBT) et le 2-mercaptobenzimidazole (MBI), sur la dissolution et la passivation à l’émergence en surface de différents types de joints de grains (JdG) ont été étudiés sur le cuivre microcristallin par voltamétrie cyclique (CV) et, in situ, à l’échelle nanométrique, par microscopie à effet tunnel électrochimique (ECSTM). En solution acide HCl(aq), MBT protège efficacement la surface des grains en bloquant la dissolution active des ions Cu(I). La couche de MBT préformée protège parfaitement les JdG CSL de bas Σ alors que les JdG aléatoires, plus réactifs, peuvent être protégés par l’accumulation de produits de réaction dans le site intergranulaire. Avec MBI, l'inhibition est moins efficace qu’avec MBT avec plus de produits de réaction Cu(I) générés sur les grains pour former un film de surface protecteur et une accumulation préférentielle locale se produisant plus fréquemment parmi les sites intergranulaires analysés. En solution alcaline NaOH(aq), les couches superficielles organiques, préformées après dissociation de l'oxyde natif en présence d’inhibiteur, bloquent la formation d'oxyde cuivreux Cu(I). L'analyse ECSTM locale révèle une réactivité intergranulaire résiduelle, sauf aux joints de macles cohérents. Aux JdG CSL et aléatoires, la formation d'oxyde Cu(I) est soit bloquée avec dissolution résiduelle et accumulation de produits de corrosion, soit incomplètement bloquée avec passivation résiduelle, en fonction de l'effet barrière de la couche organique préformée. Pour les JdG CSL de Σ élevé et les JdG aléatoires, plus réactifs, la couche organique préformée de MBT a un effet barrière plus marqué sur la formation d'oxyde Cu(I) que celle de MBI. La molécule MBT, pouvant se lier fortement à plus d'atomes de Cu que celle de MBI en raison de deux atomes de S au lieu d'un seul dans sa structure hétérocyclique, piègerait plus efficacement les atomes de Cu relâchés par la surface, offrant ainsi des propriétés de barrière améliorées à la couche organique MBT préformée, y compris dans les sites de surface les plus réactifs tels que les JdG. Ces résultats permettent de mieux comprendre les mécanismes locaux de protection conférés aux sites intergranulaires de surface par la préformation de couches moléculaires organiques d'inhibiteurs de corrosion.
Defects and interfaces are inevitably present in crystalline materials, and unveiling their effects on the mechanical behaviors of materials is of paramount significance. In this article, we reveal the mechanisms associated with interaction of defects with interfaces, and the strengthening or weakening of interfacial structures. Prominent examples include grain boundaries and twin boundaries in nanostructured polycrystalline metals, grain boundaries in two-dimensional polycrystalline graphene, and interfaces between dissimilar materials with one being atomically thin and ultra-flexible. Analysis with mechanics theory implies that mechanical properties can be tunable through interface engineering by controlling geometrical and topological features of defects and their distribution.
We study a link between the ground-state topology and the topology of the lattice via the presence of anomalous states at disclinations -- topological lattice defects that violate a rotation symmetry only locally. We first show the existence of anomalous disclination states, such as Majorana zero-modes or helical electronic states, in second-order topological phases by means of Volterra processes. Using the framework of topological crystals to construct d-dimensional crystalline topological phases with rotation and translation symmetry, we then identify all contributions to (d−2)-dimensional anomalous disclination states from weak and first-order topological phases. We perform this procedure for all Cartan symmetry classes of topological insulators and superconductors in two and three dimensions and determine whether the correspondence between bulk topology, boundary signatures, and disclination anomaly is unique.
The structure of ledges in otherwise symmetrical tilt boundaries built from atomistic simulations is investigated in copper in terms of continuous dislocation and generalized disclination fields. A ''discrete-to-continuum" crossover method is used to build the relevant kinematic and defect density fields on the basis of discrete atomic displacements appropriately defined in the boundary area. The resulting structure of incompatibility is compared with the so-called disconnection model of boundary ledges. In addition to their dislocation content, which characterizes the elastic displacement discontinuity across the boundary, the ledges appear to be characterized by discontinuities of the elastic rotation and dilatation fields, which are reflected by non-vanishing generalized disclination density fields.
Atomic structure of nonequilibrium [112] tilt grain boundaries in nickel containing disclination dipoles is studied by means of molecular dynamics simulations. Initial systems for simulations are constructed by joining together pieces of two bicrystals one of which contains a symmetric tilt GB S=11 / 62.96° and the other a GB S=105 / 57.12°, or S=125 / 55.39°, or S=31 / 52.20°, so disclination dipoles with strengths w = 5.84°, 7.58° and 10.76° are created. Stress maps plotted after relaxation at zero temperature indicate the presence of high long-range stresses induced by disclination dipoles. Excess energy of GBs due to the nonequilibrium structure is calculated. Effect of oscillating tension-compression stresses on the nonequilibrium GB structure is studied at temperature T = 300 K. The simulations show that the oscillating stress results in a generation of partial lattice dislocations by the GB, their glide across grains and sink at appropriate surfaces that results in a compensation of the disclination stress fields and recovery of an equilibrium GB structure and energy.
Accompanying the present trend of engineering systems to size reduction and design at microscopic/nanoscopic length scales, “Mechanics of dislocation fields” is aimed at describing the self-organization of dislocation ensembles at such small length scales and its consequences on the overall mechanical behavior of crystalline bodies.
The account of the fundamental interactions between the dislocations and other microscopic crystal defects is based on using smooth field quantities and the powerful tools of the mathematical theory of partial differential equations. The resulting theory is able to describe the emergence of dislocation microstructures and their evolution along complex loading paths. Scale transitions are performed between the properties of the dislocation ensembles and the mechanical behavior of the body.
Several variants of this overall scheme are examined regarding dislocation cores, electromechanical interactions of dislocations with electric charges in dielectric materials, the intermittency and scale-invariance of dislocation activity, grain-to-grain interactions in polycrystals, size effects on mechanical behavior and path dependence of strain hardening.
Disclinations, defects that accommodate rotational incompatibilities in a crystal lattice, have been described in detail in the literature, but rarely observed in solid materials. Recently, a method has been described by which it is proposed that disclination densities can be estimated using spatially resolved orientation data generated from electron backscatter diffraction or precession electron diffraction. Herein, a rigorous evaluation of this approach is performed. In this work, a series of constructed and real data sets are used to evaluate the methodology for estimating disclination densities from spatially mapped orientation data and demonstrate the inherent error associated with this approach. It is shown that the outcome of this analysis is heavily dependent on the how numerical approximations are implemented. If a self-consistent method is used, then the disclination tensor will always be zero and if an inconsistent method is used then the error in the estimation of the disclination tensor is unbounded. Therefore, although the theory behind the disclination tensor is sound, the inputs needed to calculate it are not experimentally accessible through the application of numerical approximation methods to orientation maps and a different methodology is needed.
We present a systematic first principles investigation on a group of representative low-Σ (Σ ⩽ 11) symmetric tilt grain boundaries in bcc-Fe. The grain boundary (GB) structures were constructed using both the coincident site lattice (CSL) and structural unit (SU) models. Calculations are performed to address the relation and applicability of the two models. Results suggest that on some of the GBs, the CSL and SU models may yield different atomistic structures. In these cases, their structures differ only by one vacancy, and the SU model always predicts a much lower GB formation energy than the CSL does. Further calculations on GB vacancy formation suggest that the SU model is more appropriate for describing the low-Σ bcc GBs.
We develop an explicit model for the interfacial energy in crystals that emphasizes the geometric origin of the cusps in the energy profile. We start by formulating a general class of interatomic energies that are reference-configuration-free but explicitly incorporate the lattice geometry of the ground state. In particular, away from the interface the energy is minimized by a perfect lattice. We build these attributes into the energy by locally matching, as best as possible, a perfect lattice to the atomic positions and then quantifying the local energy in terms of the inevitable remaining mismatch, hence the term lattice-matching used to describe the resulting interatomic energy. Based on this general energy, we formulate a simpler rigid-lattice model in which the atomic positions on both sides of the interface coincide with perfect, but misoriented, lattices. In addition, we restrict the lattice-matching operation to a binary choice between the perfect lattices on both sides of the interface. Finally, we prove an on the interatomic energy and use that bound as a basis for comparison with experiment. We specifically consider symmetric tilt grain boundaries (STGB), symmetric twist grain boundaries (STwGB) and asymmetric twist grain boundaries (ATwGB) in face-centered cubic (FCC) and body-centered cubic (BCC) crystals. Two or more materials are considered for each choice of crystal structure and boundary class, with the choice of materials conditioned by the availability of molecular dynamics data. Despite the approximations made, we find very good overall agreement between the predicted interfacial energy structure and that calculated by molecular dynamics. In particular, the positions of the cusps are predicted well, and therefore, although surface reconstruction and faceting are not included in the model, the dominant orientations of the facets are correctly predicted by our geometrical model.
The chapter contains sections titles:
A disclination-structural model of quasiperiodic tilt grain boundaries of finite length in polycrystalline and nanocrystalline materials is suggested. The model is exemplified by calculations of stress fields and energy characteristics of some quasiperiodic tilt grain boundaries of finite length. The results of calculations of energy characteristics are compared with analogous results obtained within the dislocation-structural model and with experimental data. The disclination description is shown to be more efficient, in particular, when considering tilt boundaries in nanocrystalline materials.
Because of the discrete nature of the dislocation structure of grain boundaries, the misorientation angle over a boundary length that is not a multiple of the distance between the structural dislocations composing the boundary differs from the misorientation angle of an infinitely long boundary. For this reason, at boundary junctions, where a boundary joins with other boundaries, geometrically necessary disclinations are formed, which compensate for the difference in misorientation. The nature of these disclinations is investigated in detail and their strength is calculated. It is shown that for a boundary length of about 10 nm the disclination strength is about 1°. The elastic energy of triple boundary junctions caused by geometrically necessary disclinations is calculated. This energy is comparable with the energy of an isolated lattice dislocation. The dependence of the energy of boundaries on the displacement of a network of grain-boundary dislocations as a rigid body with respect to junctions is considered. An increase in energy upon such displacements also is on the order of the dislocation energy. It is shown that if the grains are about 10 nm, the junction disclinations of this type can significantly affect the properties of nanocrystals.
Computer simulation using embedded-atom-method potentials was employed to study the structure of [001] tilt boundaries in nickel and copper. In both metals, the Σ= 5 (210) boundary was found to have the same stable structure known in the literature as B'.B'. The Σ = 5 (310) boundary in copper has two structures, Cand C'. In nickel, new metastable structures Cz and C'', apart from a stable structure C, were revealed. Ranges of stability of these structures in intermediate boundaries have been studied. The energies of 32 boundaries were calculated in the entire range of misorientation angles. Based on the magnitudes of the energies of favored boundaries obtained upon atomic simulation, the energy of grain boundaries was calculated in the whole range of misorientations using a disclination-structural unit model. For both metals, an almost ideal coincidence of the calculated curves of the dependence of energy on the misorientation angle with the results of simulation was obtained.
A grain boundary is described as a two-dimensional phase distinct of the 3D phase of the adjacent crystals [1]. We are now concerned with the question of the existence of structural changes specific to this two-dimensional phase and the possibility of loss of intergranular order with increasing temperature. Phase transformations at grain boundaries may also result from chemical composition variations; this effect is not considered in this section but will be discussed when dealing with the segregation phenomenon (Part II)
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
Energies for symmetric tilt grain boundaries in pure Al and in Al with substitutional Pb defects at coincident sites along the grain boundaries were calculated using a modified embedded atom method potential and density functional theory. The agreement between the analytic potential, the first principles calculations and experiment is reasonably good for the pure system. For the Al-Pb system both the analytic potential and first principles calculations predict that Pb segregation to the interface is energetically preferred compared to the dilute solution. The application of a disclination structural unit model to calculating grain boundary energies over the entire range of tilt angles is also explained.
Solute atoms in dilute alloys have been shown to segregate at grain boundaries and stabilize them against grain growth. At present, most theories of the stabilization of nanostructured alloys do not account for the detailed atomic structure of the interfaces, but instead rely on averaged segregation energies. One of the reasons for this is the daunting task of determining segregation energies for a large number of possible sites in a given microstructure. We have developed a new approach to predicting and organizing interface structures in alloys that takes advantage of perturbation techniques and a disclination structural units model (DSUM) developed previously to describe grain boundary structure and properties in pure systems. The fundamental idea is to treat dilute alloys as a perturbed form of the pure metal systems whose energy can be determined by the DSUM. This paper introduces this method and gives a preliminary validation by comparing segregation energies for zirconium solute segregating to a grain boundary in copper calculated via the perturbation method and full atomistic simulations.
The relation between two elastic continuum approaches to grain boundary structure, the dislocation and disclination models, is discussed. It is shown that the disclination model has two advantages: a well-behaved expression for the elastic energy of disclination dipole walls, which describes the elastic energy over a wide interval of misorientations, and a continuous misorientation angle dependence of the elastic energy of grain boundaries in an interval between two delimiting boundaries. The elastic energy of the most general, faceted disclination wall is calculated. For cases in which both the energies of delimiting boundaries and elastic constants are available from atomic simulations (〈001〉 and 〈111〉 tilt boundaries in copper and 〈001〉 and 〈011〉 tilt boundaries in diamond) quantitative agreement between the disclination model and simulation results is obtained.
The aim of the paper is to show experimental evidence of the rotational defects referred to as disclinations in polycrystalline aggregates. Using orientation maps obtained from electron backscattered diffraction or transmission electron microscopy, a method for the recovery of components of the disclination density tensor is presented and applied to various polycrystalline materials. Mapping the disclination densities reveals their extensive presence at intra-granular low-angle boundaries, low and high-angle grain boundaries and triple junctions, irrespective of the material symmetry and grain size. A significant level of rotational incompatibility, with dipolar distribution of the disclinations, is detected in all cases investigated. Since high-angle rotational incompatibility cannot be accounted for consistently by dislocation-based models, the present results support considering disclinations in addition to dislocations in the interpretation of grain boundaries and triple junctions.
Applications of the 0-lattice concept (which has been developed in part I) are discussed, such as the patterns of lattice points in high angle boundaries, the conservation of those patterns on a relative translation of the two lattices and dislocation networks in high angle boundaries.
The absolute interfacial energies of [001] tilt and twist boundaries in copper have been determined, by using a Zeiss Interference Microscope to measure the dihedral angles which form during the thermal grooving process. An analysis of the results obtained for small angle, lineage boundaries, in terms of the appropriate dislocation model, shows that the Read-Shockley equation predicts the energy up to misorientations of 5 to 6°, but that for larger misorientations, it gives much too small an energy. It is shown also that van der Merwe's treatment, which avoids some of the limitations present in the Read-Shockley derivation, is capable of predicting the energy up to misorientations of 8 to 9°. Moreover, from the analysis presented herein, it is possible to offer an explanation for the apparent agreement that was obtained, at large misorientations (25 to 30°), between the previous measurements of relative grain boundary energies and the Read-Shockley equation. The large angle grain boundary is characterized by a broad maximum in energy that shows no energy cusps, for either tilt or twist boundaries. Both types of boundaries have nearly the same energy for misorientations less than 18°, but for the range of misorientations, over which there is a maximum in energy, twist boundaries have a lower energy by about 130 ergs/cm2. An attempt to explain this difference from the results obtained from a calculation made on an atomic basis is discussed.
A new interpretation of the coincidence model of a grain boundary is proposed. A relative translation (without rotation) of two crystals which have a coincidence orientation relationship can lower the energy and therefore increase the stability of the boundary between them. Hence a coincidence boundary will not contain coincidence sites or shared atom sites, as was assumed by the lattice coincidence and boundary coincidence models in order to explain the different properties of coincidence and non-coincidence boundaries. It is proposed that the physical criterion for the special properties of a coincidence boundary is not coincidence per se; it is the condition for small, periodically repeating units of the structure. In boundaries that correspond exactly to a coincidence orientation all structural units are equal. Departure from the exact coincidence orientation relationship results in appropriate ‘mixing’ of units that correspond to the neighbouring coincidence orientation. The structure of non-symmetrical boundaries is discussed.
A method that has been developed to obtain the geometry of a grain boundary associated with a minimum internal energy is described in detail. A Morse potential is employed to represent the interatomic forces. The procedure was applied to a series of boundaries of the coincidence orientations. The resulting structures are consistent with experimental observations.
A coincidence model of high-angle grain boundaries can be extended to include deviations from coincidence. The generalised boundary has a terraced structure, corresponding to the densely packed planes in the coincidence lattice, and a superimposed dislocation network, corresponding to a sub-boundary in the coincidence lattice. This model is a natural extension of previous dislocation models and models based on coincidence relationships. The model explains many of the observed properties of grain boundaries and should have wide validity for the cubic system.RésuméUn modèle de coïncidence pour les joints de grains à grand angle peut être modifié de manière à inclure les écarts de coïncidence. Le joint généralisé présente une structure en terrace, correspondant aux plans denses dans le réseau de coïncidence et un réseau supplémentaire de dislocations correspondant à un sous-joint dans le réseau de coïncidence. Ce modèle est une extension naturelle des modèles de dislocations antérieurs et de modèles établis sur la base des relations de coïncidence. II explique un grand nombre des propriétés observées des joints de grains et il semblerait valable dans de nombreux cas pour le système cubique.ZusammenfassungEin Koinzidenzmodell von Groβwinkelkorngrenzen kann so erweitert werden, daβ es auch Abweichun-gen von der Koinzidenz einschlieβt. Die allgemeine Korngrenze hat eine Stufenstruktur, entsprechend den dichtest gepackten Ebenen im Koinzidenzgitter, und ein überlagertes Versetzungsnetzwerk, entsprechend einer Subkorngrenze im Koinzidenzgitter. Dieses Modell ist eine natürliche Erweiterung früherer Versetzungsmodelle und von Modellen, die auf Koinzidenzbeziehungen beruhen. Das Modell erklärt viele der beobachteten Korngrenzeneigenschaften und sollte in kubischen Systemen in vielen Fällen zutreffend sein.
Great progress in the theoretical determination of the structures and energies of grain boundary has been made possible by the advent of the computer. Here we present a method for such determinations assuming relaxation to occur. The relaxed configuration is based upon a minimum free energy criterion. Examples of the results are shown. The measured energies correlate satisfactorily with the theoretical values. The boundaries which have particularly low energies also exhibit exceptional segregation, corrosion, diffusion … characteristics.However the value of the energy proves inadequate to fully characterize the properties of a boundary, especially in the case of properties which vary with the direction in the boundary plane. The example of grain boundary sliding is considered in some detail.
Since the grain boundary is a rotational defect and so is a disclination, it is proposed here that a grain boundary may be made of disclinations instead of dislocations which are translational defects. The excess elastic energy for a symmetric tilt boundary whose orientation is between two low energy orientations is calculated by using this model. Such calculation cannot be easily performed using dislocations.
The energies and motions of grain boundaries between two crystallites are investigated theoretically using the dislocation model of grain boundaries. Quantitative predictions made for simple boundaries for cases in which the plane of the boundary contains the axis of relative rotation of the grains appear to agree with available experimental data. The quantitative expression for energy per unit area for small angles is approximately [Ga / 4Ï(1-Ï)]θ[A-lnθ] where G is the rigidity modulus, a the lattice constant, Ï Poisson's ratio, θ the relative rotation and A approximately 0.23. Grain boundaries of the form considered may permit intercrystalline slip and may act as stress raisers for the generation of dislocations.
Imperfections in Nearly Perfect Crystals
- Fisher