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A perturbative approach to the backflow dynamics of nematic defects

Published online by Cambridge University Press:  05 January 2011

PAOLO BISCARI  and
TIMOTHY J. SLUCKIN
PAOLO BISCARI
Affiliation:
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milan, Italy email: paolo.biscari@polimi.it
TIMOTHY J. SLUCKIN
Affiliation:
School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK email: t.j.sluckin@soton.ac.uk
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Abstract

We present an asymptotic theory that includes in a perturbative expansion the coupling effects between the director dynamics and the velocity field in a nematic liquid crystal. Backflow effects are most significant in the presence of defect motion, since in this case the presence of a velocity field may strongly reduce the total energy dissipation and thus increase the defect velocity. As an example, we illustrate how backflow influences the speeds of opposite-charged defects.

Keywords

Nematic liquid crystalsPerturbations, asymptoticspropagation of defects
Type
Papers
Information
European Journal of Applied Mathematics , Volume 23 , Issue 1: Liquid Crystals , February 2012 , pp. 181 - 200
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Friedel, G. (1922) (Extracts from) Les états mésomorphes de la matière (The mesomorphic states of matter). Ann. Phys. – Paris 18, 273–474.CrossRefGoogle Scholar
[2]Mermin, N. D. (1979) The topological theory of defects in ordered media. Rev. Mod. Phys. 51, 591–648.CrossRefGoogle Scholar
[3]Kléman, M. (1989) Defects in liquid crystals. Rep. Prog. Phys. 52, 555–654.CrossRefGoogle Scholar
[4]Kléman, M. & Friedel, J. (2008) Disclinations, dislocations, and continuous defects: A reappraisal. Rev. Mod. Phys. 80, 61–115.CrossRefGoogle Scholar
[5]Biscari, P. & Guidone Peroli, G. (1997) A hierarchy of defects in biaxial nematics. Commun. Math. Phys. 186, 381–392.CrossRefGoogle Scholar
[6]Oseen, C. W. (1933) The theory of liquid crystals. Trans. Faraday Soc. 29, 883–900.CrossRefGoogle Scholar
[7]Frank, F. C. (1958) On the theory of liquid crystals. Disc. Faraday Soc. 25, 19–28.CrossRefGoogle Scholar
[8]Ericksen, J. L. (1961) Hydrostatic theory of liquid crystals. Arch. Ration. Mech. Anal. 9, 371–378.CrossRefGoogle Scholar
[9]Ericksen, J. L. (1961) Anisotropic fluids. Arch. Ration. Mech. Anal. 9, 231–237.Google Scholar
[10]Ericksen, J. L. (1962) Nilpotent energies in liquid crystal theory. Arch. Ration. Mech. Anal. 10, 189–196.CrossRefGoogle Scholar
[11]Leslie, F. M. (1968) Some constitutive equations for liquid crystals. Arch. Rat. Mech. Anal. 28, 265–283.CrossRefGoogle Scholar
[12]de Gennes, P. G. (1971) Short-range order effects in the isotropic phase of nematics and cholesterics. Mol. Cryst. Liq. Cryst. 12, 193–214.CrossRefGoogle Scholar
[13]Olmsted, P. D. & Goldbartr, P. (1990) Theory of the non-equilibrium phase transition for nematic liquid crystals under shear flow. Phys. Rev. A 41, 4578–4581.CrossRefGoogle Scholar
[14]Olmsted, P. D. & Goldbart, P. (1992) Isotropic-nematic transition in shear flow: State selection, coexistence, phase transitions, and critical behaviour. Phys. Rev. A 46, 4578–4993.CrossRefGoogle Scholar
[15]Beris, A. N. & Edwards, B. J. (1994) Thermodynamics of Flowing Systems with Internal Microstructure, Oxford University Press, Oxford, UK.CrossRefGoogle Scholar
[16]Schopohl, N. & Sluckin, T. J. (1987) Defect core structure in nematic liquid crystals. Phys. Rev. Lett. 59, 2582–2584.CrossRefGoogle ScholarPubMed
[17]Volterra, V. (1905) Sull'equilibrio dei corpi elastici più volte connessi. Rend. R. Acc. Lincei 14, 193–202.Google Scholar
[18]Love, A. E. H. (1927) A Treatise on the Mathematical Theory of Elasticity, 4th ed., Cambridge University Press, Cambridge, UK.Google Scholar
[19]Burgers, J. M. (1939) Some considerations of the field of stress connected with dislocations in a regular crystal lattice. K. Ned. Akad. Wet. 42, 293–325 (Part I), 378–399 (Part II).Google Scholar
[20]Ryskin, G. & Kremenetsky, M. (1991) Drag force on a line defect moving through an otherwise undisturbed field: Disclination line in a nematic liquid crystal. Phys. Rev. Lett. 67, 1574–1577.CrossRefGoogle Scholar
[21]Denniston, C. (1996) Disclination dynamics in nematic liquid crystals. Phys. Rev. B 54, 6272–6275.CrossRefGoogle ScholarPubMed
[22]Kats, E. I., Lebedev, V. V. & Malinin, S. V. (2002) Disclination motion in liquid crystalline films. JETP 95, 714–727.CrossRefGoogle Scholar
[23]Clark, M. G. & Leslie, F. M. (1978) A calculation of orientational relaxation in nematic liquid crystals. Proc. R. Soc. Lond. A 361, 463–485.Google Scholar
[24]Pargellis, A., Turok, N. & Yurke, B. (1991) Monopole–antimonopole annihilation in a nematic liquid crystal. Phys. Rev. Lett. 67, 1570–1573.CrossRefGoogle Scholar
[25]Yakutovich, M. V., Newton, C. J. P. & Cleaver, D. J. (2009) Mesh-free simulation of complex LCD geometries. Mol. Cryst. Liq. Cryst. 502, 245–257.CrossRefGoogle Scholar
[26]Pismen, L. M. & Rubinstein, B. Y. (1992) Motion of interacting point defects in nematics. Phys. Rev. Lett. 69, 96–99.CrossRefGoogle ScholarPubMed
[27]Guidone Peroli, G. & Virga, E. G. (1996) Annihilation of point defects in nematic liquid crystals. Phys. Rev. E 54, 5235–5241.Google Scholar
[28]Guidone Peroli, G. & Virga, E. G. (1998) Dynamics of point defects in nematic liquid crystals. Physica D 111, 356–372.Google Scholar
[29]Biscari, P., Guidone Peroli, G. & Virga, E. G. (1999) A statistical study for evolving arrays of nematic point defects. Liq. Cryst. 26, 1825–1832.CrossRefGoogle Scholar
[30]Hindmarsh, M. (1995) Where are the hedgehogs in quenched nematics? Phys. Rev. Lett. 75, 2502–2505.CrossRefGoogle ScholarPubMed
[31]Richardson, G. (2000) Line disclination dynamics in uniaxial nematic liquid crystals. Q. Jl. Mech. Appl. Math. 53, 49–71.CrossRefGoogle Scholar
[32]Biscari, P. & Sluckin, T. J. (2005) Field-induced motion of nematic disclinations. SIAM J. Appl. Math. 65, 2141–2157.CrossRefGoogle Scholar
[33]Denniston, C., Orlandini, E. & Yeomans, J. M. (2000) Simulations of liquid crystal hydrodynamics in the isotropic and nematic phases. Europhys. Lett. 52, 481–487.CrossRefGoogle Scholar
[34]Denniston, C., Orlandini, E. & Yeomans, J. M. (2001) Lattice Boltzmann simulations of liquid crystal hydrodynamics. Phys. Rev. E 63, 056702.Google ScholarPubMed
[35]Tóth, G., Denniston, C. & Yeomans, J. M. (2002) Hydrodynamics of topological defects in nematic liquid crystals. Phys. Rev. Lett. 88, 105504.CrossRefGoogle ScholarPubMed
[36]Svenšek, D. & Žumer, S. (2002) Hydrodynamics of pair-annihilating disclination lines in nematic liquid crystals. Phys. Rev. E 66, 021712.Google ScholarPubMed
[37]Blanc, C., Svensek, D., Zumer, S. & Nobili, M. (2005) Dynamics of nematic liquid crystal disclinations: The role of the backflow. Phys. Rev. Lett. 95, 097802.CrossRefGoogle ScholarPubMed
[38]Grecov, D. & Rey, A. D. (2004) Impact of texture on stress growth in thermotropic liquid crystalline polymers subjected to step-shear. Rheol. Acta 44, 135–149.CrossRefGoogle Scholar
[39]Liu, C., Shen, J. & Yang, X.-F. (2007) Dynamics of defect motion in nematic liquid crystal flow: Modeling and numerical simulation. Commun. Comput. Phys. 2, 1184–1198.Google Scholar
[40]Abukhdeir, N. M. & Rey, A. D. (2008) Defect kinetics and dynamics of pattern coarsening in a two-dimensional smectic-A system. N. J. Phys. 10, 063025.CrossRefGoogle Scholar
[41]Shoarinejad, S. & Shahzamanian, M. A. (2008) On the numerical study of Frederick transition in nematic liquid crystals. J. Mol. Liq. 138, 14–19.CrossRefGoogle Scholar
[42]Kundu, S., Grecov, D., Ogale, A. A. & Rey, A. D. (2009) Shear flow-induced microstructure of a synthetic mesophase pitch. J. Rheol. 53, 85–113.CrossRefGoogle Scholar
[43]Viñals, J. (2009) Defect dynamics in mesophases. J. Phys. Soc. Jpn. 78, 041011.CrossRefGoogle Scholar
[44]Véron, A. R. & Martins, A. F. (2009) Tensorial form of Leslie–Ericksen equations and applications. Mol. Cryst. Liq. Cryst. 508, 309–336.CrossRefGoogle Scholar
[45]Sonnet, A. M. & Virga, E. G. (2009) Flow and reorientation in the dynamics of nematic defects. Liq. Cryst. 36, 1185–1192.CrossRefGoogle Scholar
[46]Leonov, A. I. (2005) On the minimum of extended dissipation in viscous nematodynamics. Rheol. Acta 44, 573–576.CrossRefGoogle Scholar
[47]de Gennes, P. G. & Prost, J. (1993) The Physics of Liquid Crystals, IInd ed., Clarendon Press, Oxford, UK.CrossRefGoogle Scholar
[48]Parodi, O. (1970) Stress tensor for a nematic liquid crystal. J. Physiq. 31, 581–584.CrossRefGoogle Scholar
[49]Stewart, I. W. (2004) The Static and Dynamic Continuum Theory of Liquid Crystals, Ist ed., Taylor & Francis, London.Google Scholar
[50]Majda, A. J. & Bertozzi, A. L. (2002) Vorticity and incompressible flow. Cambridge Texts in Applied Mathematics, Ist ed., Vol. 27, Cambridge University Press, Cambridge, UK.Google Scholar
[51]Biscari, P., Sluckin, T. J. & Turzi, S. (2011) (in preparation).Google Scholar