English

Noun

dislocations

1. Plural of dislocation

In materials science, a dislocation is a crystallographic defect, or irregularity, within a crystal structure. The presence of dislocations strongly influences many of the properties of materials. The theory was originally developed by Vito Volterra in 1905. Some types of dislocations can be visualised as being caused by the termination of a plane of atoms in the middle of a crystal. In such a case, the surrounding planes 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.

Mathematically, dislocations are a type of topological defect, sometimes called a soliton. 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), but a single dislocation typically cannot "disappear" on its own.

Dislocation geometry

There are two main types of dislocation, 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).

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 boundry. 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 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:

\sigma_ = \frac \frac

\sigma_ = \frac \frac

\tau_ = \frac \frac

where μ is the shear modulus of the material, b is the Burgers vector, ν is Poisson's ratio 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.

Dislocations, slip and plasticity

Until the 1930s, one of the enduring challenges of materials science was to explain plasticity in microscopic terms. A naive attempt to calculate the shear stress at which neighbouring atomic planes slip over each other in a perfect crystal suggests that, for a material with shear modulus G, shear strength τm is given approximately by:

\tau_m = \frac \,

As shear modulus in metals is typically within the range 20 000 to 150 000 MPa, 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 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" (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 (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 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) which promotes the recovery and subsequent recrystallisation of the material.

The combined processing techniques of work hardening and annealing allow for control over dislocation density, the degree of dislocation entanglement, and ultimately the yield strength 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

Bibliography

• [1]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 AND DEFECTS; Differential Geometry, Crystal Melting", pp. 743-1456, World Scientific (Singapore, 1989); Paperback ISBN 9971-5-0210-0 (readable online here)
• Meyers and Chawla. (1999) Mechanical Behaviors of Materials. Prentice Hall, Inc. 228-231.
• "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

External links

• 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.