Disclination

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In

line defect in which there is compensation of an angular gap. They were first discussed by Vito Volterra in 1907,[1] who have an analysis of the elastic strains of a wedge disclination. By analogy to dislocations in crystals, the term, disinclination, was first used by Frederick Charles Frank and since then has been modified to its current usage, disclination.[2] They have since been analyzed in some detail particularly by Roland deWit.[3][4]

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 fiveling nanoparticles.[5][6] 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,[3][4] with dislocation motion moving the position of a disclination.[7]

Disclinations occur in many different cases, ranging from liquid crystals[8] to nanoparticles[9][10] and in elastically distorted materials.[11]

Example in two dimensions

Formation of two disclinations (right) out of a dislocation (left) on an otherwise hexagonal background

In 2D, disclinations and

Kosterlitz–Thouless transitions
.

Equally sized discs (spheres, particles, atoms) form a hexagonal 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 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 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 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.

See also

References

  1. ISSN 0012-9593
    .
  2. ISBN 0-521-21149-2
  3. ^
    PMID 32189727
    .
  4. ^
    PMID 32189758
    .
  5. ISSN 0022-3719
    .
  6. ISSN 0141-8610
    .
  7. ISSN 0021-8979
    .
  8. ISBN 978-0-521-21149-9
    .
  9. doi:10.1002/(SICI)1521-4079(199911)34:9<1091::AID-CRAT1091>3.0.CO;2-S
    .
  10. doi:10.1016/j.matlet.2015.03.137
    .
  11. PMID 11923534
    .

Further reading
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