Topology and geometry of nematic braids

Physics Reports

Volumn 538, Issue 1, 2014, Pages 1-37

  Čopar, Simon ^a,b  

^a UNIVERSITY OF PENNSYLVANIA   (United States)

^b UNIVERSITY OF LJUBLJANA   (Slovenia)

Author keywords

Colloids; Homotopy theory; Nematic braids; Nematic liquid crystal; Self linking number; Topological defects

Indexed keywords

EID: 84897964665     PISSN: 03701573     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physrep.2014.01.001     Document Type: Review

References (136)

1. de Gennes P.G., Prost J. The Physics of Liquid Crystals 1993, Oxford University Press, Oxford.
2. Kléman M., Lavrentovich O.D. Soft Matter Physics: An Introduction 2003, Springer, New York.
3. Stephen M.J., Straley J.P. Physics of liquid crystals. Rev. Modern Phys. 1974, 45:617.
4. Kléman M. Points, Lines and Walls: In Liquid Crystals, Magnetic Systems and Various Ordered Media 1982, Wiley-Interscience Publication, Chichester.
5. Mermin N.D. The topological theory of defects in ordered media. Rev. Modern Phys. 1979, 51:591.
6. Calderon F.L., Stora T., Monval O.M., Poulin P., Bibette J. Phys. Rev. Lett. 1994, 72:2959.
7. Poulin P., Cabuil V., Weitz D.A. Phys. Rev. Lett. 1997, 79:4862.
8. Loudet J.-C., Poulin P. Phys. Rev. Lett. 2001, 87:165503.
9. Kurik M.V., Lavrentovich O.D. Defects in liquid crystals: homotopy theory and experimental studies. Usp. fiz. Nauk 1988, 154:381.
10. Vella A., Intartaglia R., Blanc C., Smalyukh I.I., Lavrentovich O.D., Nobili M. Electric-field-induced deformation dynamics of a single nematic disclination. Phys. Rev. E 2005, 71:061705.
11. Kim Y., Senyuk B., Lavrentovich D. Molecular reorientation of a nematic liquid crystal by thermal expansion. Nature Commun. 2012, 3:1133.
12. Loudet J.-C., Barois P., Poulin P. Nature 2000, 407:611.
13. Poulin P., Stark H., Lubensky T.C., Weitz D.A. Novel colloidal interactions in anisotropic fluids. Science 1997, 275:1770.
14. Lubensky T.C., Pettey D., Currier N., Stark H. Topological defects and interactions in nematic emulsions. Phys. Rev. E 1998, 57:612.
15. Stark H. Physics of colloidal dispersions in nematic liquid crystals. Phys. Rep. 2001, 351:387.
16. Gu Y., Abbott N.L. Observation of Saturn-ring defects around solid microspheres in nematic liquid crystals. Phys. Rev. Lett. 2000, 85:4719.
17. Muševič I., Škarabot M., Tkalec U., Ravnik M., Žumer S. Two-dimensional nematic colloidal crystals self-assembled by topological defects. Science 2006, 313:954.
18. Tkalec U., Muševič I. Topology of nematic liquid crystal colloids confined to two dimensions. Soft Matter 2013, 9:8140.
19. Lapointe C.P., Mason T.G., Smalyukh I.I. Shape-controlled colloidal interactions in nematic liquid crystals. Science 2009, 326:1083.
20. Dontabhaktuni J., Ravnik M., Žumer S. Shape-tuning the colloidal assemblies in nematic liquid crystals. Soft Matter 2012, 8:1657.
21. Senyuk B., Liu Q., He S., Kamien R.D., Kusner R.B., Lubensky T.C., Smalyukh I.I. Topol. colloids 2013.
22. Liu Q., Senyuk B., Tasinkevych M., Smalyukh I.I. Nematic liquid crystal boojums with handles on colloidal handlebodies. Proc. Natl. Acad. Sci. USA 2013, 110:9231.
23. Muševič I., Škarabot M., Babič D., Osterman N., Poberaj I., Nazarenko V., Nych A. Laser trapping of small colloidal particles in a nematic liquid crystal: clouds and ghosts. Phys. Rev. Lett. 2004, 93:187801.
24. Smalyukh I.I., Senyuk B.I., Shiyanovskii S.V., Lavrentovich O.D., Kuzmin A.N., Kachynski A.V., Prasad P.N. Optical trapping, manipulaton and 3D imaging of disclinations in liquid crystals and measurement of their line tension. Mol. Cryst. Liq. Cryst. 2006, 450:79.
25. Lee T., Trivedi R.P., Smalyukh I.I. Multimodal nonlinear optical polarizing microscopy of long-range molecular order in liquid crystals. Opt. Lett. 2010, 35:3447.
26. Smalyukh I.I., Kachynski A.V., Kuzmin A.N., Prasad P.N. Laser trapping in anisotropic fluids and polarization-controlled particle dynamics. Proc. Natl. Acad. Sci. USA 2006, 103:18048.
27. Trivedi R.P., Engström D., Smalyukh I.I. Optical manipulation of colloids and defect structures in anisotropic liquid crystal fluids. J. Opt. 2011, 13:044001.
28. Araki T., Tanaka H. Colloidal aggregation in a nematic liquid crystal: topological arrest of particles by a single-stroke disclination line. Phys. Rev. Lett. 2006, 97:127801.
29. Ognysta U., Nych A., Nazarenko V., Škarabot M., Muševič I. Design of 2D binary colloidal crystals in a nematic liquid crystal. Langmuir 2009, 25:12092.
30. Ravnik M., Škarabot M., Žumer S., Tkalec U., Poberaj I., Babič D., Osterman N., Muševič I. Entangled nematic colloidal dimers and wires. Phys. Rev. Lett. 2007, 99:247801.
31. Tkalec U., Ravnik M., Čopar S., Žumer S., Muševič I. Reconfigurable knots and links in chiral nematic colloids. Science 2011, 333:62.
32. Hijnen N., Wood T.A., Wilson D., Clegg P.S. Self-organization of particles with planar surface anchoring in a cholesteric liquid crystal. Langmuir 2010, 26:13502.
33. Nych A., Ognysta U., Škarabot M., Ravnik M., Žumer S., Muševič I. Assembly and control of 3d nematic dipolar colloidal crystals. Nature Commun. 2013, 4:1489.
34. Care C.M., Cleaver D.J. Computer simulation of liquid crystals. Rep. Progr. Phys. 2005, 68:2665.
35. Wilson M.R., Allen M.P. Computer simulations of mesogenic molecules using realistic atom-atom potentials. Mol. Cryst. Liq. Cryst. 1991, 198:465.
36. Berardi R., Muccioli L., Zannoni C. Can nematic transitions be predicted by atomistic simulations? A computational study of the odd-even effect. Chem. Phys. Chem. 2004, 5:104.
37. Berardi R., Emerson A.P.J., Zannoni C. Monte Carlo investigation of a Gay-Berne liquid crystal. J. Chem. Soc. Faraday Trans. 1993, 89:4069.
38. Lebwohl P.A., Lasher G. Nematic-liquid-crystal order-a Monte Carlo calculation. Phys. Rev. A 1972, 6:426.
39. Fabbri U., Zannoni C. A Monte Carlo investigation of the Lebwohl-Lasher lattice model in the vicinity of its orientational phase transition. Mol. Phys. 1986, 58:763.
40. Kralj S., Virga E.G., Žumer S. Biaxial torus around nematic point defects. Phys. Rev. E 1999, 60:1858.
41. Schopohl N., Sluckin T.J. Defect core structure in nematic liquid crystals. Phys. Rev. Lett. 1987, 59:2582.
42. Kléman M., Lavrentovich O.D. Topological point defects in nematic liquid crystals. Phil. Mag. 2006, 86:4117.
43. Frank F.C. I. liquid crystals. on the theory of liquid crystals. Discuss. Faraday Soc. 1958, 25:19.
44. Chandrasekhar S. Liquid Crystals 1992, Cambridge University Press. second ed.
45. Blanc C., Svenšek D., Žumer S., Nobili M. Dynamics of nematic liquid crystal disclinations: the role of the backflow. Phys. Rev. Lett. 2005, 95:097802.
46. Huang H.-W. Hydrodynamics of nematic liquid crystals. Phys. Rev. Lett. 1971, 26:1525.
47. Denniston C., Marenduzzo D., Orlandini E., Yeomans J.M. Lattice boltzmann algorithm for three-dimensional liquid-crystal hydrodynamics. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 2004, 362:1745.
48. Tóth G., Denniston C., Yeomans J.M. Hydrodynamics of domain growth in nematic liquid crystals. Phys. Rev. E 2003, 67:051705.
49. Tóth G., Denniston C., Yeomans J.M. Hydrodynamics of topological defects in nematic liquid crystals. Phys. Rev. Lett. 2002, 88:105504.
50. Svenšek D., Žumer S. Hydrodynamics of pair-annihilating disclination lines in nematic liquid crystals. Phys. Rev. E 2002, 66:021712.
51. Fletcher C.A.J. Computational Techniques for Fluid Dynamics 1 1991, Springer-Verlag, Berlin.
52. Fukuda J., Yoneya M., Yokoyama H. Defect structure of a nematic liquid crystal around a spherical particle: adaptive mesh refinement approach. Phys. Rev. E 2002, 65:041709.
53. Fukuda J., Stark H., Yoneya M., Yokoyama H. Interaction between two spherical particles in a nematic liquid crystal. Phys. Rev. E 2004, 69:041706.
54. Guzmán O., Kim E.B., Grollau S., Abbott N.L., de Pablo J.J. Defect structure around two colloids in a liquid crystal. Phys. Rev. Lett. 2003, 91:235507.
55. Kim E.B., Guzmán O., Grollau S., Abbott N.L., de Pablo J.J. Interactions between spherical colloids mediated by a liquid crystal: a molecular simulation and mesoscale study. J. Chem. Phys. 2004, 121:1949.
56. Ravnik M., Žumer S. Nematic braids: 2D entangled nematic liquid crystal colloids. Soft Matter 2009, 5:4520.
57. Ravnik M., Žumer S. Nematic colloids entangled by topological defects. Soft Matter 2009, 5:269.
58. Škarabot M., Ravnik M., Žumer S., Tkalec U., Poberaj I., Babič D., Osterman N., Muševič I. Interactions of quadrupolar nematic colloids. Phys. Rev. E 2008, 77:031705.
59. Tanaka H., Araki T. Simulation method of colloidal suspensions with hydrodynamic interactions: fluid particle dynamics. Phys. Rev. Lett. 2000, 85:1338.
60. Tkalec U., Ravnik M., Žumer S., Muševič I. Vortexlike topological defects in nematic colloids: chiral colloidal dimers and 2D crystals. Phys. Rev. Lett. 2009, 103:127801.
61. Čopar S., Porenta T., Žumer S. Nematic disclinations as twisted ribbons. Phys. Rev. E 2011, 84:051702.
62. Ravnik M., Alexander G.P., Yeomans J.M., Žumer S. Mesoscopic modelling of colloids in chiral nematics. Faraday Discuss. 2010, 144:159.
63. Ravnik M., Žumer S. Landau-de Gennes modelling of nematic liquid crystal colloids. Liq. Cryst. 2009, 36:1201.
64. Mackay F.E., Denniston C. Modelling defect-bonded chains produced by colloidal particles in a cholesteric liquid crystal. EPL 2011, 94:66003.
65. Nobili M., Durand G. Disorientation-induced disordering at a nematic-liquid-crystal-solid interface. Phys. Rev. A 1992, 46:6174.
66. Fournier J.-B., Galatola P. Modeling planar degenerate wetting and anchoring in nematic liquid crystals. EPL 2005, 72:403.
67. Čopar S., Porenta T., Jampani V.S.R., Muševič I., Žumer S. Stability and rewiring of nematic braids in chiral nematic colloids. Soft Matter 2012, 8:8595.
68. Čopar S., Porenta T., Žumer S. Visualisation methods for complex nematic fields. Liq. Cryst. 2013, 40:1759.
69. Dubois-Violette E., Parodi O. Émulsions nématiques. Effets de champ magnétiques et effets piézoélectriques. J. Phys. Paris 1969, 30:4.
70. Biscari P., Peroli G.G. A hierarchy of defects in biaxial nematics. Comm. Math. Phys. 1997, 186:381.
71. Saupe A. Disclinations and properties of the directorfield in nematic and cholesteric liquid crystals. Mol. Cryst. Liq. Cryst. 1973, 21:211.
72. Nash C., Sen S. Topology and Geometry for Physicists 1983, Academic Press, Inc., New York.
73. Trebin H.R. The topology of non-uniform media in condensed matter physics. Adv. Phys. 1982, 31:195.
74. Toulouse G., Kléman M. Principles of a classification of defects in ordered media. J. Phys. Paris 1976, 37:149.
75. Toulouse G., Michel L., Kléman M. Classification of topologically stable defects in ordered media. J. Phys. Lett. 1977, 38:195.
76. Kléman M., Michel L. Spontaneous breaking of Euclidean invariance and classification of topologically stable defects and configurations of crystals and liquid crystals. Phys. Rev. Lett. 1978, 40:1387.
77. Dzyaloshinskii I.E. Domains and dislocations in antiferromagnets. JETP Lett. 1977, 25:98.
78. Alexander G.P., Chen B.G., Matsumoto E.A., Kamien R.D. Disclination loops, point defects, and all that in nematic liquid crystals. Rev. Modern Phys. 2012, 84:497.
79. B.G. Chen, Topological defects in nematic and smectic liquid crystals, Ph.D. Thesis, University of Pennsylvania, 2012.
80. Chen B.G., Ackerman P., Alexander G.P., Kamien R.D., Smalyukh I.I. Threading the needle: generating structures in nematics. Phys. Rev. Lett. 2013, 110:237801.
81. Jänich K. Topological properties of ordinary nematics in 3-space. Acta Appl. Math. 1987, 8:65.
82. Nakanishi H., Hayashi K., Mori H. Topological classification of unknotted ring defects. Comm. Math. Phys. 1988, 117:203.
83. Monastyrsky M.I., Retakh V.S. Topology of linked defects in condensed matter. Comm. Math. Phys. 1986, 103:445.
84. Smalyukh I.I., Lansac Y., Clark N.A., Trivedi R.P. Three-dimensional structure and multistable optical switching of triple-twisted particle-like excitations in anisotropic fluids. Nature Mater. 2010, 9:139.
85. Fukuda J. Continuous transformation of a -1/2 wedge disclination line to a +1/2 one. Phys. Rev. E 2010, 81:040701.
86. Fukuda J., Yoneya M., Yokoyama H. Simulation of cholesteric blue phases using a Landau-de Gennes theory: effect of an applied electric field. Phys. Rev. E 2009, 80:031706.
87. Fukuda J., Žumer S. Ring defects in a strongly confined chiral liquid crystal. Phys. Rev. Lett. 2011, 106:097801.
88. Fukuda J., Žumer S. Quasi-two-dimensional Skyrmion lattices in a chiral nematic liquid crystal. Nature Commun. 2011, 2:246.
89. Chuang I., Yurke B., Pargellis A.N. Coarsening dynamics in uniaxial nematic liquid crystals. Phys. Rev. E 1993, 47:3343.
90. Terentjev E.M. Disclination loops, standing alone and around solid particles in nematic liquid crystals. Phys. Rev. E 1995, 51:1330.
91. Ruhwandl R.W., Stark H. Long-range forces and aggregation of colloid particles in a nematic liquid crystal. Phys. Rev. E 1997, 55:2958.
92. Poulin P., Weitz D.A. Inverted and multiple nematic emulsions. Phys. Rev. E 1998, 57:626.
93. Škarabot M., Ravnik M., Žumer S., Tkalec U., Poberaj I., Babič D., Muševič I. Hierarchical self-assembly of nematic colloidal superstructures. Phys. Rev. E 2008, 77:061706.
94. Busch K., John S. Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum. Phys. Rev. Lett. 1999, 83:967.
95. Kang D., Maclennan J.E., Clark N.A., Zakhidov A.A., Baughman R.H. Electro-optic behavior of liquid-crystal-filled silica opal photonic crystals: effect of liquid-crystal allignment. Phys. Rev. Lett. 2001, 86:4052.
96. Takeda H., Yoshino K. Tunable photonic band schemes of opals and inverse opals infiltrated with liquid crystals. J. Appl. Phys. 2002, 92:8658.
97. Mach P., Wiltzius P., Megens M., Weitz D.A., Lin K., Lubensky T.C., Yodh A.G. Switchable Bragg diffraction from liquid crystal in colloid-templated structures. Europhys. Lett. 2002, 58:679.
98. Trivedi R.P., Klevets I.I., Senyuk B., Lee T., Smalyukh I.I. Reconfigurable interactions and three-dimensional patterning of colloidal particles and defects in lamellar soft media. Proc. Natl. Acad. Sci. USA 2012, 109:4744.
99. Yu-Sheng W., Guo-Hong Y., Li-Jun T., Yi-Shi D. Stability of disclinations in nematic liquid crystals. Commun. Theor. Phys. 2006, 45:993.
100. Yada M., Yamamoto J., Yokoyama H. Direct observation of anisotropic interparticle forces in nematic colloids with optical tweezers. Phys. Rev. Lett. 2004, 92:185501.
101. Smalyukh I.I., Kuzmin A.N., Kachynski A.V., Prasad P.N., Lavrentovich O.D. Optical trapping of colloidal particles and measurement of the defect line tension and colloidal forces in a thermotropic nematic liquid crystal. Appl. Phys. Lett. 2005, 86:021913.
102. Jampani V.S.R., Škarabot M., Ravnik M., Čopar S., Žumer S., Muševič I. Colloidal entanglement in highly twisted chiral nematic colloids: twisted loops, Hopf links, and trefoil knots. Phys. Rev. E 2011, 84:031703.
103. Čopar S., Žumer S. Nematic braids: topological invariants and rewiring of disclinations. Phys. Rev. Lett. 2011, 106:177801.
104. Ishikawa T., Lavrentovich O.D. Crossing of disclinations in nematic slabs. Europhys. Lett. 1998, 41:171.
105. Čopar S., Clark N.A., Ravnik M., Žumer S. Elementary building blocks of nematic disclination networks in densely packed 3d colloidal lattices. Soft Matter 2013, 9:8203.
106. Kamien R.D. The geometry of soft materials: a primer. Rev. Mod. Phys. 2002, 74:953.
107. Dennis M.R., Hannay J.H. Geometry of Cǎlugǎreanu's theorem. Proc. R. Soc. A 2005, 461:3245.
108. White J.H. Self-linking and the Gauss integral in higher dimensions. Amer. J. Math. 1969, 91:693.
109. Pohl W.F. The self-linking number of a closed space curve. J. Mat. Mech. 1968, 17:975.
110. Rögen P. An index formula for the self-linking number of a space curve. Geom. Dedicata 2008, 134:197.
111. Fuller F.B. Decomposition of the linking number of a closed ribbon: a problem from molecular biology. Proc. Natl. Acad. Sci. USA 1978, 75:3557.
112. Fuller F.B. The writhing number of a space curve. Proc. Nat. Acad. Sci. U.S.A. 1971, 86:815.
113. Kamien R.D. Local writhing dynamics. Eur. Phys. J. B 1998, 1:1.
114. Calvo J., Millett K., Rawdon E., Stasiak A. On the writhing number of a non-closed curve. Physical and Numerical Models in Knot Theory Including Applications to the Life Sciences 2005, 525-545. World Scientific Publishing, Singapore.
115. Sumners D.W.L., Ricca R.L., Moffatt H.K., Khesin B., Kauffman L.H., Berger M.A. Topological quantities: calculating winding, writhing, linking, and higher order invariants. Lecture Notes in Mathematics 2009, vol. 1973:75-97. Springer, Berlin, Heidelberg.
116. Berger M.A., Prior C. The writhe of open and closed curves. J. Phys. A: Math. Gen. 2006, 39:8321.
117. Guillemin V., Pollack A. Differential Topology 1974, Prentice-Hall, Inc., Englewood Cliffs, New Jersey.
118. Cantarella J. On comparing the writhe of a smooth curve to the writhe of an inscribed polygon. Numer. Anal. 2005, 42:1846.
119. Samuel J., Sinha S. Tops and writhing DNA. J. Stat. Phys. 2011, 143:399.
120. S. Kutter, Elasticity of polymers with internal topological constraints, Ph.D. Thesis, University of Cambridge, 2002.
121. Avelino P.P., Moraes F., Oliveira J.C.R.E., de Oliveira B.F. Electric field induced inversion of the sign of half-integer disclinations in nematic liquid crystals. Soft Matter 2011, 7:10961.
122. Čopar S., Žumer S. Quaternions and hybrid nematic disclinations. Proc. R. Soc. A 2013, 469:2156.
123. Wright D.C., Mermin N.D. Crystalline liquids: the blue phases. Rev. Modern Phys. 1989, 61:385.
124. Čopar S., Dennis M.R., Kamien R.D., Žumer S. Singular values, nematic disclinations and emergent biaxiality. Phys. Rev. E 2013, 87:050504.
125. 3 in liquid crystals by the homotopic topology methods. Sov. Phys.-JETP 1977, 45:1186.
126. Meyer R.B. On the existence of even indexed disclinations in nematic liquid crystals. Phil. Mag. 1972, 72:405.
127. Cladis P.E., Kléman M. Non-singular disclinations of strength s=+1 in nematics. J. Phys. Paris 1972, 33:591.
128. Poenaru V., Toulouse G. The crossing of defects in ordered media and the topology of 3-manifolds. J. Phys. 1977, 38:887.
129. Kléman M. Relationship between burgers circuit, volterra process and homotopy groups. J. Phys. France Lett. 1977, 38:199.
130. Volovik G.E. Topological singularities on the surface of an ordered system. JETP Lett. 1978, 28:59.
131. Poénaru V. Some aspects of the theory of defects of ordered media and gauge fields related to foliations. Comm. Math. Phys. 1981, 80:127.
132. Chen B.G., Alexander G.P., Kamien R.D. Symmetry breaking in smectics and surface models of their singularities. Proc. Natl. Acad. Sci. 2009, 106:15577.
133. Biscari P., Napoli G., Turzi S. Bulk and surface biaxiality in nematic liquid crystals. Phys. Rev. E 2006, 74:031708.
134. Seč D., Porenta T., Ravnik M., Žumer S. Geometrical frustration of chiral ordering in cholesteric droplets. Soft Matter 2012, 8:11982.
135. Kurik M.V., Lavrentovich O.D. Topological defects of cholesteric liquid crystals for volumes of spherical shape. Mol. Cryst. Liq. Cryst. 1982, 72:239.
136. M. Ravnik, Colloidal structures confined to thin nematic layers, Ph.D. Thesis, University of Ljubljana, 2009.