Disclination

In crystallography, a disclination is a line defect in which there is compensation of an angular gap. They were first discussed by Vito Volterra in 1907,[1] who provided 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 materials, 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 dislocations 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 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

  • Hexatic phase – two-dimensional state of matter characterized by two order parameters: a short-range positional and a quasi-long-range orientational (sixfold) order
  • Fiveling – Five crystals arranged round a common axis

References

  1. ^ Volterra, Vito (1907). "Sur l'équilibre des corps élastiques multiplement connexes". Annales scientifiques de l'École normale supérieure. 24: 401–517. doi:10.24033/asens.583. ISSN 0012-9593.
  2. ^ Chandrasekhar, S. (1977) Liquid Crystals, Cambridge University Press, p. 123, ISBN 0-521-21149-2
  3. ^ a b deWit, Roland (1973). "Theory of disclinations: II. Continuous and discrete disclinations in anisotropic elasticity" (PDF). Journal of Research of the National Bureau of Standards, Section A. 77A (1): 49–100. doi:10.6028/jres.077A.003. ISSN 0022-4332. PMC 6742835. PMID 32189727.
  4. ^ a b deWit, Roland (1973). "Theory of disclinations: IV. Straight disclinations". Journal of Research of the National Bureau of Standards, Section A. 77A (5): 607–658. doi:10.6028/jres.077a.036. ISSN 0022-4332. PMC 6728463. PMID 32189758.
  5. ^ deWit, Roland (1972). "Partial disclinations". Journal of Physics C: Solid State Physics. 5 (5): 529–534. Bibcode:1972JPhC....5..529D. doi:10.1088/0022-3719/5/5/004. ISSN 0022-3719.
  6. ^ Howie, A.; Marks, L. D. (1984). "Elastic strains and the energy balance for multiply twinned particles". Philosophical Magazine A. 49 (1): 95–109. Bibcode:1984PMagA..49...95H. doi:10.1080/01418618408233432. ISSN 0141-8610.
  7. ^ deWit, Roland (1971). "Relation between Dislocations and Disclinations". Journal of Applied Physics. 42 (9): 3304–3308. Bibcode:1971JAP....42.3304D. doi:10.1063/1.1660730. ISSN 0021-8979.
  8. ^ Chandrasekhar, S. (1977). Liquid crystals. Cambridge monographs on physics. Cambridge ; New York: Cambridge University Press. ISBN 978-0-521-21149-9.
  9. ^ Gryaznov, V. G.; Heydenreich, J.; Kaprelov, A. M.; Nepijko, S. A.; Romanov, A. E.; Urban, J. (1999). "Pentagonal Symmetry and Disclinations in Small Particles". Crystal Research and Technology. 34 (9): 1091–1119. Bibcode:1999CryRT..34.1091G. doi:10.1002/(SICI)1521-4079(199911)34:9<1091::AID-CRAT1091>3.0.CO;2-S.
  10. ^ Ji, Wenhai; Qi, Weihong; Li, Xu; Zhao, Shilei; Tang, Shasha; Peng, Hongcheng; Li, Siqi (2015). "Investigation of disclinations in Marks decahedral Pd nanoparticles by aberration-corrected HRTEM". Materials Letters. 152: 283–286. Bibcode:2015MatL..152..283J. doi:10.1016/j.matlet.2015.03.137.
  11. ^ Murayama, M.; Howe, J. M.; Hidaka, H.; Takaki, S. (2002). "Atomic-Level Observation of Disclination Dipoles in Mechanically Milled, Nanocrystalline Fe". Science. 295 (5564): 2433–2435. Bibcode:2002Sci...295.2433M. doi:10.1126/science.1067430. ISSN 0036-8075. PMID 11923534.

Further reading