crystallography Crystallography is the branch of science devoted to the study of molecular and crystalline structure and properties. The word ''crystallography'' is derived from the Ancient Greek word (; "clear ice, rock-crystal"), and (; "to write"). In J ...

, a disclination is a

line defect A crystallographic defect is an interruption of the regular patterns of arrangement of atoms or molecules in crystalline solids. The positions and orientations of particles, which are repeating at fixed distances determined by the unit cell para ...

in which there is compensation of an angular gap. They were first discussed by

Vito Volterra Vito Volterra (, ; 3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to Mathematical and theoretical biology, mathematical biology and Integral equation, integral equations, being one of the ...

in 1907, who provided an analysis of the elastic strains of a wedge disclination. By analogy to

dislocation In materials science, a dislocation or Taylor's dislocation is a linear crystallographic defect or irregularity within a crystal structure that contains an abrupt change in the arrangement of atoms. The movement of dislocations allow atoms to sli ...

s in crystals, the term, ''disinclination'', was first used by

Charles Frank Charles Reser Frank (born April 17, 1947) is an American actor noted for playing Bret Maverick's cousin Ben Maverick in the 1978 TV movie ''The New Maverick'' with James Garner and Jack Kelly, and in the short-lived 1979 television series '' ...

and since then has been modified to its current usage, ''disclination''. They have since been analyzed in some detail particularly by Roland deWit. Disclinations are characterized by an angular vector (called a Frank vector), and the line of the disclination. When the vector and the line are the same they are sometimes called ''wedge disclinations'' which are common in decahedral nanoparticles. When the Frank vector and the line of the disclination are at right angles they are called ''twist disclinations''. As pointed out by John D. Eshelby there is an intricate connection between disclinations and dislocations, with dislocation motion moving the position of a disclination. Disclinations occur in many different materials, ranging from

liquid crystal Liquid crystal (LC) is a state of matter whose properties are between those of conventional liquids and those of solid crystals. For example, a liquid crystal can flow like a liquid, but its molecules may be oriented in a common direction as i ...

s to decahedral and

icosahedral In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...

nanoparticles, and in elastically distorted materials.

Example in two dimensions

In 2D, disclinations and

s are point defects instead of line defects as in 3D. They are

topological defect In mathematics and physics, solitons, topological solitons and topological defects are three closely related ideas, all of which signify structures in a physical system that are stable against perturbations. Solitons do not decay, dissipate, dispe ...

s and play a central role in melting of 2D crystals within the

KTHNY theory In statistical mechanics, the Kosterlitz–Thouless–Halperin–Nelson–Young (KTHNY) theory describes the process of melting of crystals in two dimensions (2D). The name is derived from the initials of the surnames of John Michael Kosterlitz, ...

, based on two Kosterlitz–Thouless transitions. Equally sized discs (spheres, particles, atoms) form a

hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A regular hexagon is de ...

al crystal as dense packing in two dimensions. In such a crystal, each particle has six nearest neighbors. Local strain and twist (for example induced by thermal motion) can cause configurations where discs (or particles) have a

coordination number In chemistry, crystallography, and materials science, the coordination number, also called ligancy, of a central atom in a molecule or crystal is the number of atoms, molecules or ions bonded to it. The ion/molecule/atom surrounding the central ion ...

different of six, typically five or seven. Disclinations are topological defects, therefore (starting from a hexagonal array) they can only be created in pairs. Ignoring surface/border effects, this implies that there are always as many 5-folded as 7-folded disclinations present in a perfectly plane 2D crystal. A "bound" pair of 5-7-folded disclinations is a dislocation. If myriad dislocations are thermally dissociated into isolated disclinations, then the monolayer of particles becomes an

isotropic In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also ...

fluid in two dimensions. A 2D crystal is free of disclinations. To transform a section of a hexagonal array into a 5-folded disclination (colored green in the figure), a triangular wedge of hexagonal elements (blue triangle) has to be removed; to create a 7-folded disclination (orange), an identical wedge must be inserted. The figure illustrates how disclinations destroy orientational order, while dislocations only destroy translational order in the far field (portions of the crystal far from the center of the disclination). Disclinations are topological defects because they cannot be created locally by an

affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, '' affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More general ...

without cutting the hexagonal array outwards to infinity (or the border of a finite crystal). The undisturbed hexagonal crystal has a 60° symmetry, but when a wedge is removed to create a 5-folded disclination, the crystal symmetry is stretched to 72° – for a 7-folded disclination, it is compressed to about 51,4°. Thus, disclinations store elastic energy by disturbing the director field.

Example in two dimensions

See also

References

Further reading