In materials science Materials science is an interdisciplinary field involving the properties of matter and its applications to various areas of science and engineering. This science investigates the relationship between the structure of materials at atomic or molecular scales and their macroscopic properties. It includes elements of applied physics and chemistry, a dislocation is a crystallographic defect Crystalline solids have a very regular atomic structure: that is, the local positions of atoms with respect to each other are repeated at the atomic scale. These arrangements are called crystal structures, and their study is called crystallography. However, most crystalline materials are not perfect: the regular pattern of atomic arrangement is or irregularity, within a crystal structure In mineralogy and crystallography, crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry. Patterns are located upon the points of a lattice, which is an array. The presence of dislocations strongly influences many of the properties of materials. The theory was originally developed by Vito Volterra Vito Volterra was an Italian mathematician and physicist, best known for his contributions to mathematical biology in 1905. Some types of dislocations can be visualized as being caused by the termination of a plane of atoms The atom is a basic unit of matter that consists of a dense, central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons . The electrons of an atom are bound to the nucleus by the electromagnetic force. Likewise, a group of atoms can remain in the middle of a crystal A crystal or crystalline solid is a solid material, whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is crystallography. The process of crystal formation via mechanisms of crystal growth is called. In such a case, the surrounding planes In mathematics, a plane is any flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line (one-dimension) and a space (three-dimensions). Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of are not straight, but instead bend around the edge of the terminating plane so that the crystal structure is perfectly ordered on either side. The analogy with a stack of paper is apt: if a half a piece of paper is inserted in a stack of paper, the defect in the stack is only noticeable at the edge of the half sheet.
There are two primary types: edge dislocations and screw dislocations. Mixed dislocations are intermediate between these.
Figure 1: An edge-dislocation (b = Burgers vector The Burgers vector, named after Dutch physicist Jan Burgers, is a vector, often denoted b, that represents the magnitude and direction of the lattice distortion of dislocation in a crystal lattice)Mathematically, dislocations are a type of topological defect In mathematics and physics, a topological soliton or a topological defect is a solution of a system of partial differential equations or of a quantum field theory homotopically distinct from the vacuum solution; it can be proven to exist because the boundary conditions entail the existence of homotopically distinct solutions. Typically, this, sometimes called a soliton In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. (The term "dispersive effects" refers to a property of certain systems where the speed of the waves varies. The mathematical theory explains why dislocations behave as stable particles: they can be moved about, but maintain their identity as they move. Two dislocations of opposite orientation, when brought together, can cancel each other (this is the process of annihilation Annihilation is defined as "total destruction" or "complete obliteration" of an object; having its root in the Latin nihil . A literal translation is "to make into nothing"), but a single dislocation typically cannot "disappear" on its own.
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Dislocation geometry
Figure A Crystal lattice showing atoms and lattice planesTwo main types of dislocation exist: edge and screw. Dislocations found in real materials typically are mixed, meaning that they have characteristics of both.
A crystalline material consists of a regular array of atoms, arranged into lattice planes (imagine stacking oranges in a grocers, each of the trays of oranges are the lattice planes). One approach is to begin by considering a 3-d representation of a perfect crystal lattice, with the atoms represented by spheres. The viewer may then start to simplify the representation by visualising planes of atoms instead of the atoms themselves (Figure A).
Figure B Schematic diagram (lattice planes) showing an edge dislocation. Burgers vector in black, dislocation line in blue.Edge dislocations
An edge dislocation is a defect where an extra half-plane of atoms is introduced mid way through the crystal, distorting nearby planes of atoms. When enough force is applied from one side of the crystal structure, this extra plane passes through planes of atoms breaking and joining bonds with them until it reaches the grain boundary. A simple schematic diagram of such atomic planes can be used to illustrate lattice defects such as dislocations. (Figure B represents the "extra half-plane" concept of an edge type dislocation). The dislocation has two properties, a line direction, which is the direction running along the bottom of the extra half plane, and the Burgers vector The Burgers vector, named after Dutch physicist Jan Burgers, is a vector, often denoted b, that represents the magnitude and direction of the lattice distortion of dislocation in a crystal lattice which describes the magnitude and direction of distortion to the lattice. In an edge dislocation, the Burgers vector is perpendicular to the line direction.
The stresses caused by an edge dislocation are complex due to its inherent asymmetry. These stresses are described by three equations:[1]
where μ is the shear modulus The shear modulus is concerned with the deformation of a solid when it experiences a force parallel to one of its surfaces while its opposite face experiences an opposing force . In the case of an object that's shaped like a rectangular prism, it will deform into a parallelepiped. Anisotropic materials such as wood and paper exhibit differing of the material, b is the Burgers vector The Burgers vector, named after Dutch physicist Jan Burgers, is a vector, often denoted b, that represents the magnitude and direction of the lattice distortion of dislocation in a crystal lattice, ν is Poisson's ratio Poisson's ratio , named after Siméon Poisson, is the ratio, when a sample object is stretched, of the contraction or transverse strain (perpendicular to the applied load), to the extension or axial strain (in the direction of the applied load) and x and y are coordinates.
These equations suggest a vertically oriented dumbbell of stresses surrounding the dislocation, with compression experienced by the atoms near the "extra" plane, and tension experienced by those atoms near the "missing" plane.[1]
Screw dislocations
Top right: edge dislocation. Bottom right: screw dislocation. Figure C Schematic diagram (lattice planes) showing a screw dislocation.A screw dislocation is much harder to visualize. Imagine cutting a crystal along a plane and slipping one half across the other by a lattice vector, the halves will fit back together without leaving a defect. If the cut only goes part way through the crystal, and then slipped, the boundary of the cut is a screw dislocation. It comprises a structure in which a helical A helix is a type of space curve, i.e. a smooth curve in three-dimensional space. It is characterised by the fact that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helixes are coil springs and the handrails of spiral staircases. A "filled-in" helix – for example, a spiral ramp – path is traced around the linear defect (dislocation line) by the atomic planes in the crystal lattice (Figure C). Perhaps the closest analogy is a spiral-sliced ham. In pure screw dislocations, the Burgers vector is parallel to the line direction.
Despite the difficulty in visualization, the stresses caused by a screw dislocation are less complex than those of an edge dislocation. These stresses need only one equation, as symmetry allows only one radial coordinate to be used:[1]
where μ is the shear modulus The shear modulus is concerned with the deformation of a solid when it experiences a force parallel to one of its surfaces while its opposite face experiences an opposing force . In the case of an object that's shaped like a rectangular prism, it will deform into a parallelepiped. Anisotropic materials such as wood and paper exhibit differing of the material, b is the Burgers vector, and r is a radial coordinate. This equation suggests a long cylinder of stress radiating outward from the cylinder and decreasing with distance. Please note, this simple model results in an infinite value for the core of the dislocation at r=0 and so it is only valid for stresses outside of the core of the dislocation.[1]
Mixed dislocations
In many materials, dislocations are found where the line direction and Burgers vector are neither perpendicular nor parallel and these dislocations are called mixed dislocations, consisting of both screw and edge character.
Observation of dislocations
Transmission Electron Micrograph of DislocationsWhen a dislocation line intersects the surface of a metallic material, the associated strain field locally increases the relative susceptibility of the material to acidic etching In industry, etching, also known as chemical milling, is the process of using acids, bases or other chemicals to dissolve unwanted materials such as metals, semiconductor materials or glass. This process has been used on a wide variety of metals with depths of metal removal as large as 12mm . Selective attack by the chemical reagent on different and an etch pit of regular geometrical format results. If the material is strained (deformed) and repeatedly re-etched, a series of etch pits can be produced which effectively trace the movement of the dislocation in question.
Transmission electron microscopy Transmission electron microscopy is a microscopy technique whereby a beam of electrons is transmitted through an ultra thin specimen, interacting with the specimen as it passes through. An image is formed from the interaction of the electrons transmitted through the specimen; the image is magnified and focused onto an imaging device, such as a can be used to observe dislocations within the microstructure Microstructure is defined as the structure of a prepared surface or thin foil of material as revealed by a microscope above 25× magnification. The microstructure of a material can strongly influence physical properties such as strength, toughness, ductility, hardness, corrosion resistance, high/low temperature behavior, wear resistance, and so on, of the material. Thin foils of metallic samples are prepared to render them transparent to the electron beam of the microscope. The electron The electron is a subatomic particle carrying a negative electric charge. It has no known components or substructure, and therefore is believed to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton. The intrinsic angular momentum of the electron is a half integer value in units of ħ, which means that beam suffers diffraction Diffraction refers to various phenomena which occur when a wave encounters an obstacle. It is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings. Similar effects are observed when light waves travel through a medium with a varying refractive index or a sound wave through one with by the regular crystal lattice planes of the metal atoms and the differing relative angles between the beam and the lattice planes of each grain in the metal's microstructure result in image contrast (between grains of different crystallographic orientation). The less regular atomic structures of the grain boundaries A grain boundary is the interface between two grains in a polycrystalline material. Grain boundaries disrupt the motion of dislocations through a material, so reducing crystallite size is a common way to improve strength, as described by the Hall-Petch relationship. Since grain boundaries are defects in the crystal structure they tend to decrease and in the strain fields around dislocation lines have different diffractive properties than the regular lattice within the grains, and therefore present different contrast effects in the electron micrographs. (The dislocations are seen as dark lines in the lighter, central region of the micrographs on the right). Transmission electron micrographs of dislocations typically utilize magnifications of 50,000 to 300,000 times (though the equipment itself offers a wider range of magnifications than this).
Transmission Electron Micrograph of DislocationsSome microscopes also permit the in-situ heating and/or deformation of samples, thereby permitting the direct observation of dislocation movement and their interactions. Note the characteristic 'wiggly' contrast of the dislocation lines as they pass through the thickness of the material. Note also that a dislocation cannot end within a crystal; the dislocation lines in these images end at the sample surface. A dislocation can only be contained within a crystal as a complete loop.
Field ion microscopy and atom probe The atom probe is an atomic-resolution microscope used in materials science that was invented in 1967 by Erwin Wilhelm Müller, J. A. Panitz, and S. Brooks McLane techniques offer methods of producing much higher magnifications (typically 3 million times and above) and permit the observation of dislocations at an atomic level. Where surface relief can be resolved to the level of an atomic step, screw dislocations appear as distinctive spiral features - thus revealing an important mechanism of crystal growth: where there is a surface step, atoms can more easily add to the crystal, and the surface step associated with a screw dislocation is never destroyed no matter how many atoms are added to it.
(By contrast, traditional optical microscopy The optical microscope, often referred to as the "light microscope", is a type of microscope which uses visible light and a system of lenses to magnify images of small samples. Optical microscopes are the oldest and simplest of the microscopes. Digital microscopes are now available which use a CCD camera to examine a sample, and the, which is not appropriate for the direct observation of dislocations, typically offers magnifications up to a maximum of only around 2000 times).
After chemical etching, small pits are formed where the etching solution preferentially attacks the sample surface around the dislocations intercepting this surface, due to the more highly strained state of the material . Thus, the image features indicate points at which dislocations intercept the sample surface. In this way, dislocations in silicon, for example, can be observed indirectly using an interference microscope. Crystal orientation can be determined by the shape of the etch pits associated with the dislocations (in the case of the illustration below; 100 elliptical, 111 - triangular/pyramidal).
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Dislocations in silicon, orientation 100 |
Dislocations in silicon, orientation 111 |
Dislocation in silicon, orientation 111 |
Sources of dislocations
Dislocation density in a material can be increased by plastic deformation by the following relationship: . Since the dislocation density increases with plastic deformation, a mechanism for the creation of dislocations must be activated in the material. Three mechanisms for dislocation formation are formed by homogeneous nucleation, grain boundary initiation, and interfaces the lattice and the surface, precipitates, dispersed phases, or reinforcing fibers.
The creation of a dislocation by homogeneous nucleation is a result of the rupture of the atomic bonds along a line in the lattice. A plane in the lattice is sheared, resulting in 2 oppositely faced half planes or dislocations. These dislocations move away from each other through the lattice. Since homogeneous nucleation forms dislocations from perfect crystals and requires the simultaneous breaking of many bonds, the energy required for homogeneous nucleation is high. For instance the stress required for homogeneous nucleation in copper has been shown to be , where G is the shear modulus of copper (46 GPa). Solving for , we see that the required stress is 3.4 GPa, which is very close to the theoretical strength of the crystal. Therefore, in conventional deformation homogeneous nucleation requires a concentrated stress, and is very unlikely. Grain boundary initiation and interface interaction are more common sources of dislocations.
Irregularities at the grain boundaries in materials can produce dislocations which propagate into the grain. The steps and ledges at the grain boundary are an important source of dislocations in the early stages of plastic deformation.
The surface of a crystal can produce dislocations in the crystal. Due to the small steps on the surface of most crystals, stress in certain regions on the surface is much larger than the average stress in the lattice. The dislocations are then propagated into the lattice in the same manner as in grain boundary initiation. In monocrystals, the majority of dislocations are formed at the surface. The dislocation density 200 micrometres into the surface of a material has been shown to be six times higher than the density in the bulk. However, in polycrystalline materials the surface sources cannot have a major effect because most grains are not in contact with the surface.
The interface between a metal and an oxide can greatly increase the number of dislocations created. The oxide layer puts the surface of the metal in tension because the oxygen atoms squeeze into the lattice, and the oxygen atoms are under compression. This greatly increases the stress on the surface of the metal and consequently the amount of dislocations formed at the surface. The increased amount of stress on the surface steps results in an increase of dislocations.[2]
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Dislocation source (schematic) |
Stresses in a dislocation pile-up |
Dislocations, slip and plasticity
Main article: Dislocation creepUntil the 1930s, one of the enduring challenges of materials science was to explain plasticity In physics and materials science, plasticity describes the deformation of a material undergoing non-reversible changes of shape in response to applied forces. For example, a solid piece of metal or plastic being bent or pounded into a new shape displays plasticity as permanent changes occur within the material itself. In engineering, the in microscopic terms. A naive attempt to calculate the shear stress A shear stress, denoted , is defined as a stress which is applied parallel or tangential to a face of a material, as opposed to a normal stress which is applied perpendicularly at which neighbouring atomic planes slip over each other in a perfect crystal suggests that, for a material with shear modulus The shear modulus is concerned with the deformation of a solid when it experiences a force parallel to one of its surfaces while its opposite face experiences an opposing force . In the case of an object that's shaped like a rectangular prism, it will deform into a parallelepiped. Anisotropic materials such as wood and paper exhibit differing G, shear strength τm is given approximately by:
As shear modulus in metals A metal is a chemical element that is a good conductor of both electricity and heat and forms cations and ionic bonds with non-metals. In chemistry, a metal is an element, compound, or alloy characterized by high electrical conductivity. In a metal, atoms readily lose electrons to form positive ions (cations). Those ions are surrounded by is typically within the range 20 000 to 150 000 MPa The pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength. It is a measure of force per unit area, defined as one newton per square metre. In everyday life, the pascal is perhaps best known from meteorological barometric pressure reports, where it occurs in the form of hectopascals (1 hPa ≡ 100, this is difficult to reconcile with shear stresses in the range 0.5 to 10 MPa observed to produce plastic deformation in experiments.
In 1934, Egon Orowan, Michael Polanyi Michael Polanyi, FRS (March 11, 1891, Budapest – February 22, 1976, Northampton, England) was a Hungarian–British polymath whose thought and work extended across physical chemistry, economics, and philosophy. He was a Fellow of the Royal Society and a Fellow of Merton College, Oxford and G. I. Taylor, roughly simultaneously, realized that plastic deformation could be explained in terms of the theory of dislocations. Dislocations can move if the atoms from one of the surrounding planes break their bonds and rebond with the atoms at the terminating edge. In effect, a half plane of atoms is moved in response to shear stress by breaking and reforming a line of bonds, one (or a few) at a time. The energy required to break a single bond is far less than that required to break all the bonds on an entire plane of atoms at once. Even this simple model of the force required to move a dislocation shows that plasticity is possible at much lower stresses than in a perfect crystal. In many materials, particularly ductile materials, dislocations are the "carrier" of plastic deformation, and the energy required to move them is less than the energy required to fracture the material. Dislocations give rise to the characteristic malleability of metals.
When metals are subjected to "cold working Work hardening, also known as strain hardening, is the strengthening of a metal by plastic deformation. This strengthening occurs because of dislocation movements within the crystal structure of the material. Any material with a reasonably high melting point such as metals and alloys can be strengthened in this fashion[citation needed]. Alloys not" (deformation at temperatures which are relatively low as compared to the material's absolute melting temperature, Tm, i.e., typically less than 0.3 Tm) the dislocation density increases due to the formation of new dislocations and dislocation multiplication. The consequent increasing overlap between the strain fields of adjacent dislocations gradually increases the resistance to further dislocation motion. This causes a hardening of the metal as deformation progresses. This effect is known as strain hardening Work hardening, strain hardening, or cold work is the strengthening of a material by, macroscopically speaking, plastic deformation . As the material becomes increasingly saturated with new dislocations, more dislocations are prevented from nucleating (a resistance to dislocation-formation develops). This resistance to dislocation-formation (also “work hardening”). Tangles of dislocations are found at the early stage of deformation and appear as non well-defined boundaries; the process of dynamic recovery Recovery is a process by which deformed grains can reduce their stored energy by the removal or rearrangement of defects in their crystal structure. These defects, primarily dislocations, are introduced by plastic deformation of the material and act to increase the yield strength of a material. Since recovery reduces the dislocation density the leads eventually to the formation of a cellular structure containing boundaries with misorientation lower than 15° (low angle grain boundaries). In addition, adding pinning points that inhibit the motion of dislocations, such as alloying elements, can introduce stress fields that ultimately strengthen the material by requiring a higher applied stress to overcome the pinning stress and continue dislocation motion.
The effects of strain hardening by accumulation of dislocations and the grain structure formed at high strain can be removed by appropriate heat treatment (annealing Annealing, in metallurgy and materials science, is a heat treatment wherein a material is altered, causing changes in its properties such as strength and hardness. It is a process that produces conditions by heating to above the re-crystallization temperature and maintaining a suitable temperature, and then cooling. Annealing is used to induce) which promotes the recovery Recovery is a process by which deformed grains can reduce their stored energy by the removal or rearrangement of defects in their crystal structure. These defects, primarily dislocations, are introduced by plastic deformation of the material and act to increase the yield strength of a material. Since recovery reduces the dislocation density the and subsequent recrystallisation of the material.
The combined processing techniques of work hardening Work hardening, also known as strain hardening, is the strengthening of a material by, macroscopically speaking, plastic deformation . As the material becomes increasingly saturated with new dislocations, more dislocations are prevented from nucleating (a resistance to dislocation-formation develops). This resistance to dislocation-formation and annealing allow for control over dislocation density, the degree of dislocation entanglement, and ultimately the yield strength The yield strength or yield point of a material is defined in engineering and materials science as the stress at which a material begins to deform plastically. Prior to the yield point the material will deform elastically and will return to its original shape when the applied stress is removed. Once the yield point is passed some fraction of the of the material.
Dislocation climb
Dislocations can slip in planes containing both the dislocation and the Burgers Vector. For a screw dislocation, the dislocation and the Burgers vector are parallel, so the dislocation may slip in any plane containing the dislocation. For an edge dislocation, the dislocation and the Burgers vector are perpendicular, so there is only one plane in which the dislocation can slip. There is an alternative mechanism of dislocation motion, fundamentally different from slip, that allows an edge dislocation to move out of its slip plane, known as dislocation climb. Dislocation climb allows an edge dislocation to move perpendicular to its slip plane.
The driving force for dislocation climb is the movement of vacancies through a crystal lattice. If a vacancy moves next to the boundary of the extra half plane of atoms that forms an edge dislocation, the atom in the half plane closest to the vacancy can "jump" and fill the vacancy. This atom shift "moves" the vacancy in line with the half plane of atoms, causing a shift, or positive climb, of the dislocation. The process of a vacancy being absorbed at the boundary of a half plane of atoms, rather than created, is known as negative climb. Since dislocation climb results from individual atoms "jumping" into vacancies, climb occurs in single atom diameter increments.
During positive climb, the crystal shrinks in the direction perpendicular to the extra half plane of atoms because atoms are being removed from the half plane. Since negative climb involves an addition of atoms to the half plane, the crystal grows in the direction perpendicular to the half plane. Therefore, compressive stress in the direction perpendicular to the half plane promotes positive climb, while tensile stress promotes negative climb. This is one main difference between slip and climb, since slip is caused by only shear stress.
One additional difference between dislocation slip and climb is the temperature dependence. Climb occurs much more rapidly at high temperatures than low temperatures due to an increase in vacancy motion. Slip, on the other hand, has only a small dependence on temperature.
Notes
- ^ a b c d Reed-Hill, R. E. (1994) "Physical Metallurgy Principles" ISBN 0-534-92173-6
- ^ Meyers and Chawla. (1999) Mechanical Behaviors of Materials. Prentice Hall, Inc. 228-231.
References
- Reed-Hill, R. E. (1994) "Physical Metallurgy Principles" ISBN 0-534-92173-6
- Dieter, G. E. (1986) Mechanical Metallurgy ISBN 0-07-100406-8
- Honeycombe, R.W.K. (1984) The Plastic Deformation of Metals ISBN 0-7131-2181-5
- Hull, D. & Bacon, D. J. (1984) Introduction to Dislocations ISBN 0-08-028720-4
- Read, W. T. Jr. (1953) Dislocations in Crystals ISBN 1-114-49066-0
- Kleinert, Hagen, Gauge Fields in Condensed Matter, Vol. II, "STRESSES In continuum mechanics, stress is a measure of the average force per unit area of a surface within a deformable body on which internal forces act. In other words, it is a measure of the intensity of the internal forces acting between particles of a deformable body across imaginary internal surfaces . These internal forces are produced between the AND DEFECTS; Differential Geometry Differential geometry is a mathematical discipline that uses the methods of differential and integral calculus, as well as linear and multilinear algebra, to study problems in geometry. The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis for its initial development in the eighteenth and, Crystal Melting Melting, or fusion, is a physical process that results in the phase change of a substance from a solid to a liquid. The internal energy of a substance is increased, typically by the application of heat or pressure, resulting in a rise of its temperature to the melting point, at which the rigid ordering of molecular entities in the solid breaks", pp. 743-1456, World Scientific (Singapore, 1989); Paperback ISBN 9971-5-0210-0 (readable online here)
- "Atomistically-informed Dislocation Dynamics in fcc Crystals", E. Martinez, J. Marian, A. Arsenlis, M. Victoria, J. M. Perlado, Journal of the Mechanics and Physics of Solids, Volume 56, Issue 3, March 2008, Pages 869-895
- Markenscoff, Xanthippi; Anurag Gupta (2006). Collected Works of J.D. Eshelby, the Mechanics of Defects and Inhomogeneities. Springer. ISBN The International Standard Book Number is a unique numeric commercial book identifier based upon the 9-digit Standard Book Numbering (SBN) code created by Gordon Foster, now Emeritus Professor of Statistics at Trinity College, Dublin, for the booksellers and stationers W.H. Smith and others in 1966 1-4020-4416-X.
External links
| Wikimedia Commons has media related to: Dislocations in crystals |
- Defects in Crystals/ Prof. Dr. Helmut Föll website Chapter 5 contains a wealth of information on dislocations;
- DoITPoMS Online tutorial on dislocations, including movies of dislocations in bubble rafts;
- Scanning Tunneling Microscope - Gallery Image gallery, including a dislocations page, seen at the atomic level of metal surfaces, by the surface physics group at the Faculty of Physics, Vienna University of Technology, Austria.
Categories: Crystallographic defects
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