Dislocation

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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 real 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.

Figure 1: An edge-dislocation (b = Burgers vector)

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. While two dislocations of opposite orientation, when brought together, can cancel each other (this is the process of annealing), there is no way a single dislocation can "disappear" on its own.

1 Dislocation geometry
- 1.1 Edge dislocations
- 1.2 Screw dislocations
2 Observation of Dislocations
3 Dislocations, slip and plasticity
4 Bibliography
5 External links

[edit] Dislocation geometry

A dislocation can be visualized by imagining 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. But if the cut only goes part way though the crystal, the boundary of the cut will leave a defect, distorting the nearby lattice. This boundary is the line of the dislocation; the direction of the slip is the Burgers vector.

Dislocations are labeled by the angle between the dislocation line and the Burgers vector. The special cases of 90° and 0° are known as edge and screw dislocations. The dislocations present in real crystalline solids are generally mixed rather than edge or screw; the actual angles of dislocations depend on the lattice structure.

The Burgers vector for an edge dislocation is marked in black in Figure D. It is perpendicular to the dislocation line (marked in blue in Figure D) in the case of the edge, and parallel to it in the case of the screw. In metallic materials, b is aligned with close-packed crystallographic directions and its magnitude is equivalent to one interatomic spacing.

[edit] Edge dislocations

Figure A Perfect (simple cubic) crystal lattice of atoms

Figure C Simplified Representation of Lattice planes

Figure B Crystal lattice showing atom planes

Alternatively, edge dislocations can be visualised as being formed by adding an extra half-plane of atoms to a perfect crystal, so that a defect is created in the regular crystal structure along the line where the extra half-plane ends (Figure 1). Such visualisations can be difficult to interpret. Initially, it can be helpful to follow the process of simplification involved in arriving at such representations.One approach is to begin by considering a 3-d representation of a perfect crystal lattice, with the atoms represented by spheres (Figure A). The viewer may then start to simplify the representation by visualising planes of atoms instead of the atoms themselves (Figures B and C).

Figure D Schematic diagram (lattice planes) showing an edge dislocation. Burgers vector in black, dislocation line in blue.

Finally a simple schematic diagram of such atomic planes can be used to illustrate lattice defects such as dislocations. (Figure D represents the "extra half-plane" concept of an edge type dislocation).

[edit] Screw dislocations

Figure E Schematic diagram (lattice planes) showing a screw dislocation.

Screw dislocations are more difficult to visualize, but can be considered as being formed by the insertion of a "parking garage ramp" that extends to the "edges of the garage" into an otherwise perfectly layered structure. Basically it comprises a structure in which a helical path is traced around the linear defect (dislocation line) by the atomic planes in the crystal lattice (Figure E).

[edit] Observation of Dislocations

Transmission Electron Micrograph of Dislocations

When 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 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 can be used to observe dislocations within the microstructure of the material. Thin foils of metallic samples are prepared to render them transparent to the electron beam of the microscope. The electron beam suffers diffraction 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 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). Some 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 charcteristic '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 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, which is not appropriate for the 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 more highly strained material around the dislocations. 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 dislocations - 100 elliptical, 111 - triangular (pyramidal).

Dislocations in silicon, orientation 100

Dislocations in silicon, orientation 111

Dislocation in silicon, orientation 111

[edit] 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 τ_m is given approximately by:

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. Even a simple model of the force required to move a dislocation shows that shear is possible at much lower stresses than in a perfect crystal. (Hence, 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, T_m, i.e., typically less than 0.3 T_m) 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 15º (low angle grain boundaries).

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.

[edit] Bibliography

Dieter, G. E. (1988) 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)