crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...

, a disclination is a line defect in which rotational symmetry is violated. In analogy with dislocations in crystals, the term, ''disinclination'', for

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 may flow like a liquid, but its molecules may be oriented in a crystal-like way. T ...

s first used by

Frederick Charles Frank Sir Frederick Charles Frank, OBE, FRS (6 March 1911 – 5 April 1998) was a British theoretical physicist. He is best known for his work on crystal dislocations, including (with Thornton Read) the idea of the Frank–Read source of dislocation ...

and since then has been modified to its current usage, ''disclination''. It is a defect in the orientation of

director Director may refer to: Literature * ''Director'' (magazine), a British magazine * ''The Director'' (novel), a 1971 novel by Henry Denker * ''The Director'' (play), a 2000 play by Nancy Hasty Music * Director (band), an Irish rock band * ''D ...

whereas a

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 s ...

is a defect in positional order.

Example in two dimensions

In 2D, disclinations and

s are point defects instead of line defects as in 3D. They are topological defects and play a central role in melting of 2D crystals within the KTHNY theory, based on two Kosterlitz–Thouless transitions. Equally sized discs (spheres, particles, atoms) form a

hexagonal 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'' ha ...

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 io ...

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 Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...

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 generall ...

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