Dislocations In Quasicrystals Research Articles

Mechanics of dislocations and metadislocations in quasicrystals and their approximants: power invariance and balance

We discuss the mechanics of a single dislocation in quasicrystals. We account for dislocation’s core by means of a non-constant line energy. Above all, we focus attention on the origin of the balance equations involving the pertinent bulk and line actions. We derive all balances, including those of configurational actions along the dislocation line, from a unique invariance requirement for the so-called relative power; it is an axiom that furnishes also the action–reaction principle for the standard traction and the phason one, as well as the standard Cauchy theorem and, in addition, the existence of phason stress and self-action. The theoretical scheme presented here includes the description of the large core of metadislocations in quasicrystal’s approximants (complex metallic alloys). Each metadislocation features thousands of atoms per unit thickness in the direct vicinity of the core, a circumstance suggesting to consider it as a sort of micro-rod, a view adopted in the present paper. In fact, from an abstract point of view, the results presented here describe the motion of a “rod” within a body with vectorial microstructure.

Renormalized energy for dislocations in quasi-crystals

Anti-plane shear deformations of a hexagonal quasi-crystal with multiple screw dislocations are considered. Using a variational formulation, the elastic equilibrium is characterized via limit of minimizers of a core-regularized energy functional. A sharp estimate of the asymptotic energy when the core radius tends to zero is obtained using higher-order Γ-convergence. Also, the interaction between dislocations and the Peach–Köhler force at each dislocation are analyzed.

Eshelbian mechanics of novel materials: Quasicrystals

In this work, the so-called Eshelbian or configurational mechanics of quasicrystals is presented. Quasicrystals are considered as a prototype of novel materials. Material balance laws for quasicrystalline materials with dislocations are derived in the framework of generalized incompatible elasticity theory of quasicrystals. Translations, scaling transformations as well as rotations are examined; the latter presents particular interest due to the quasicrystalline structure. This derivation provides important quantities of the Eshelbian mechanics, as the Eshelby stress tensor, the scaling flux vector, the angular momentum tensor, the configurational forces (Peach–Koehler force, Cherepanov force, inhomogeneity force or Eshelby force), the configurational work, and the configurational vector moments for dislocations in quasicrystals. The corresponding [Formula: see text]-, [Formula: see text]-, and [Formula: see text]-integrals for dislocation loops and straight dislocations in quasicrystals are derived and discussed. Moreover, the explicit formulas of the [Formula: see text]-, [Formula: see text]-, and [Formula: see text]-integrals for parallel screw dislocations in one-dimensional hexagonal quasicrystals are obtained. Through this derivation, the physical interpretation of the [Formula: see text]-, [Formula: see text]-, and [Formula: see text]-integrals for dislocations in quasicrystals is revealed and their connection to the Peach–Koehler force, the interaction energy and the rotational vector moment (torque) of dislocations in quasicrystals is established.

Quasicrystals. A Primer, second edition. By Christian Janot. Oxford Classic Texts in the Physical Sciences. Oxford University Press, 2012. Pp. 427. Price GBP 42.50 (paperback). ISBN 978-0-19-965740-7.

This book on quasicrystals by C. Janot was first published in 1992, with a second edition in 1994. This version of the book is the same as the second edition, but has a foreword by Dan Shechtman, who received the Nobel Prize in Chemistry in 2011. The book is written for a non-specialist audience in a very pedagogical way. After a general introduction on How to fill space with atoms in condensed matter states, the second chapter presents the different ways to synthesize and characterize quasicrystals. The third chapter gives an introduction to high-dimensional (or superspace) crystallography, starting with simple one-dimensional quasicrystals, with a generalization then to three-dimensional quasicrystals. The fourth chapter, Where are the atoms?, presents a few examples of structural analysis in which the high-dimensional approach is used. The fifth chapter deals with the dynamics of quasicrystals and phonons, but also introduces the notion of phasons and the concept of dislocations in quasicrystals. The last chapter presents a discussion of the difficult questions of how the quasicystalline long-range order propagates and whether it is possible with finite-range interactions. This chapter ends with a presentation of the electronic properties of quasicrystals. Each chapter ends with a few exercises, which are very useful for testing the knowledge gained from the chapter. Although the material in the book has not been updated for 20 years, this book remains a classic introduction to quasicrystals and will be very useful for an undergraduate or a curious scientist intrigued by the quasicrystalline world.

On the equations of motion of dislocations in quasicrystals

In this work we investigate the theory of dynamics of dislocations in quasicrystals. We consider three models: the elastodynamic model of wave type, the elasto-hydrodynamic model, and the elastodynamic model of wave-telegraph type. Similarities and differences between the three models are pointed out and discussed. Using the framework of linear incompatible elastodynamic theory, the equations of motion of dislocations are deduced for these three models. Especially, the equations of motion for the phonon and phason elastic distortion tensors and elastic velocity vectors are derived, where the source fields are given in terms of the phonon and phason dislocation density and dislocation current tensors in analogy to the classical theory of elastodynamics of dislocations. The equations of motion for the displacement fields are also obtained.

Elasticity and dislocations in quasicrystals with 18-fold symmetry

Abstract Elasticity problems of quasicrystals with 18-fold rotational symmetry are studied. Constitutive equations and governing equations are obtained. For static elasticity problems, the displacement vectors in two phason fields are expressed in terms of two pairs of associated harmonic functions or two analytic functions. For dynamic problems, the displacement vectors can be represented in terms of an auxiliary function satisfying a fourth-order partial differential equation. A general solution of phasons is given by the solution of two diffusion equations. Phason elastic fields induced by a dislocation in a quasicrystal with 18-fold symmetry are determined and exhibit an inverse singularity.

Dislocations in icosahedral quasicrystals

Dislocations in quasicrystals, as a direct result of the lack of translational symmetry in these materials, possess various salient features. The Burgers vector of a dislocation in an icosahedral quasicrystal is a 6-dimensional vector, which reflects the fact that the dislocation, besides the phonon-type strain field analogous to dislocations in ordinary crystals, is associated inseparably with a further type of defect, the phasons. Phasons are critically involved in the formation and motion of dislocations in quasicrystals and govern the macroscopic plastic behaviour of these materials. In this article the properties of dislocations in icosahedral quasicrystals are comprehensively reviewed, starting from a continuum-mechanical description, via core-structure simulation, to their full experimental characterization. The experimental results presented address the icosahedral phases in the well explored systems Al-Pd-Mn and Zn-Mg-Dy.

Steady-state propagation of dislocations in quasi-crystals

We analyse the steady propagation at constant speed, lower than the shear wave-speed, of a straight dislocation in an unbounded elastic quasi-crystal with five-fold symmetry. We discuss only the ideal elastic behaviour, neglecting the dissipation associated with the atomic rearrangements. Under these conditions, we provide the expressions of phonon and phason fields in closed form. Both phonon and phason stresses appear to be singular near to the dislocation core. We also find the explicit expression of the energy per unit length around a moving dislocation.

Generalized dynamics of moving dislocations in quasicrystals

A theoretical framework for dislocation dynamics in quasicrystals is provided according tothe continuum theory of dislocations. Firstly, we present the fundamental theory formoving dislocations in quasicrystals giving the dislocation density tensors and introducingthe dislocation current tensors for the phonon and phason fields, including the Bianchiidentities. Next, we give the equations of motion for the incompatible elastodynamics aswell as for the incompatible elasto-hydrodynamics of quasicrystals. We continue withthe derivation of the balance law of pseudomomentum thereby obtaining thegeneralized forms of the Eshelby stress tensor, the pseudomomentum vector, thedynamical Peach–Koehler force density and the Cherepanov force density forquasicrystals. The form of the dynamical Peach–Koehler force for a straight dislocationis obtained as well. Moreover, we deduce the balance law of energy that givesrise to the generalized forms of the field intensity vector and the elastic powerdensity of quasicrystals. The above balance laws are produced for both models.The differences between the two models and their consequences are revealed.The influences of the phason fields as well as of the dynamical terms are alsodiscussed.

HRTEM observation of displacement fields around dislocations in quasicrystals

HRTEM observation of displacement fields around dislocations in quasicrystals

Walk of a Line Defect in Quasicrystals

Quasicrystalline alloys are quasiperiodic aggregates in which local atomic changes (phason activity) influence prominently the gross deformative behavior. The effects of these rearrangements on a line defect are analyzed here. Balances of standard and phason interactions are derived along the line together with the balance of actions driving eventually it the former are found to be covariant while the latter follows from a requirement of invariance with respect to the action of the group of isochoric diffeomorphisms. The results can be applied to the analysis of dislocations in quasicrystals.

Evaluation of some Fourier integrals inN-dimensional space for the theory of dislocations in quasicrystals

A method to evaluate some Fourier integrals is extended from two-dimensional (2-D) and three dimensional (3-D) spaces ton-dimensional (n-D) space, which are often used in the elasticity theory of dislocations in quasicrystals. Some key formulae have been given.

Dislocations in quasicrystals

Quasicrystals have additional elastic degrees of freedom not found in conventional crystals, which are termed phason degrees of freedom. Dislocations in quasicrystals are special in that they are inevitably accompanied by the phason strain in addition to the conventional elastic strain. Characteristic features of such quasicrystalline dislocations and plastic properties of quasicrystals are reviewed. Deformation mechanisms in quasicrystals by dislocation motion are discussed.

Dislocation kinetics in quasicrystals under small applied stress

The sliding of dislocations in quasicrystals under small applied stress is considered within phenomenological model. Implications of the quasi-periodic Peierls potential for the dislocation pinning is analyzed and associated activation mechanism is shown to be relevant for the plastic deformation of quasicrystals. The dependence of the dislocation velocity and strain rate on the applied stress is estimated. Kinetics of the dislocation movement in the quasi-periodic pinning potential is analyzed in exactly solvable one-dimensional model. The spectrum of the dislocation jump lengths is found to be a universal function of applied stress. The critical values of resolved stress leading to the drastic changes in the dislocation velocity are shown to form periodical sequence in a logarithm scale. Recent experiments on the mechanical properties of quasicrystals are discussed within proposed model.

Defocus convergent beam electron diffraction determination of Burgers vectors of dislocations in quasicrystals

Principles, method and some application examples of determining Burgers vectors of dislocations in crystals and quasicrystals by means of defocus convergent-beam electron diffraction (CBED) technique are described and reviewed and compared with contrast experiment techniques. By using defocus CBED technique, dislocation reactions during high-temperature plastic deformation of face-centered icosahedral quasicrystals have been studied. These studies lead to a preliminary understanding to the micromechanism of high-temperature plastic deformation of quasicrystals.

A Straight Dislocation in One-Dimensional Hexagonal Quasicrystals

The elastic field induced by a straight dislocation in a one-dimensional hexagonal quasicrystal with its line parallel to the quasiperiodic axis is obtained by superposition of the elastic fields of a pure edge part and a pure screw part. By extending the Peach-Koehler force to quasicrystals, the force between two parallel screw dislocations is given. For a screw dislocation in this quasicrystal of general cross-section, the elastic field and the image force on the dislocation are given via the conformal mapping method.

Dislocations in quasicrystals and their interaction with cluster-like obstacles

A dislocation moving through a quasicrystal leaves in its wake a fault denoted a phason wall. For a two-dimensional model quasicrystal the disregistry energy of this phason wall is studied to determine possible Burgers vectors of the quasicrystalline structure. Unlike periodic crystals, the disregistry energy is an average quantity with large fluctuations on the atomic scale. Therefore the dislocation core structure and mobility cannot be linked to this quantity, e.g. by a Peierls-Nabarro model. Atomistic simulations show that dislocation motion is controlled by local obstacles inherent to the atomic structure of the quasicrystal.

Energy characteristics of edge dislocations in quasicrystals

A model is developed for describing phason defects in quasicrystals in the form of dilation filaments. This model is used to calculate the energy of edge dislocations in quasicrystals including the interaction of this type of dislocation with its “intrinsic” phason defects and with the equilibrium phason defects present in a quasicrystal. It is shown that the contribution of “intrinsic” phason defects to the total energy of an edge dislocation in a quasicrystal is substantial.

Evaluation of some useful integrals in the theory of dislocations in quasicrystals

A method based on mathematical induction and Green's functions is presented for the evaluation of some integrals used often in the elasticity theory of dislocations in quasicrystals.

Disvections: mismatches, dislocations, and non-Abelian properties of quasicrystals

Complete dislocations in quasicrystals are intersections of dislocations in a high-dimensional lattice with an irrational cut that represents the physical space. We study the properties which proceed from that definition, either by methods relevant to the Volterra process, or by topological methods. The Volterra process applies in the high-dimensional lattice in a rather trivial way, but its restriction to the quasicrystal introduces very unusual geometrical properties, described in terms of phason deformations and of their singularities (mismatches). For example, the motion of a defect is generically non-commutative: the `landscape' of mismatches carried by complete dislocations or disclinations in motion depends on the path which is followed by the defect between two positions, when two such paths surround a defect. The same type of argument applies, mutatis mutandis, to the question of the intersection of two defects. Other properties, of a more metallurgical nature, ensue from that use of the Volterra process, like the existence of stacking faults in QCs, bound by incomplete dislocations, and the relationship between mismatches and the reshuffling of atoms. Now, if one wishes to describe these remarkable properties with the sole use of physical observables (i.e. without mentioning the high-dimensional lattice), it appears that the natural language is the language of the topological theory of defects in the quasilattice. In particular it is shown that the group which classifies the dislocations is non-Abelian, a property in a direct relationship with the above-mentioned non-commutativity. We give the name of disvections to complete dislocations, because of their relationship with Cartan's transvections, which are translations with non-Abelian characters in a hyperbolic space.