Revisiting the Two-Dimensional Defect-Free Azimuthal Nematic Equilibrium on an Annulus
Abstract
We study the azimuthal defect-free nematic state on a two-dimensional annulus within a simplified and reduced two-dimensional Landau--de Gennes model for nematic liquid crystals. We perform a detailed asymptotic analysis of the instabilities of the defect-free state in terms of a dimensionless material and temperature-dependent variable and the annular aspect ratio. The asymptotic analysis is accompanied by a rigorous local stability result, again in terms of a dimensionless material and temperature-dependent parameter and annular aspect ratio. In contrast to Oseen--Frank predictions, the defect-free state can be unstable in this model, with elastic isotropy and strong anchoring, for a range of macroscopically relevant annular aspect ratios.
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References
1.
J. Alvarado, Biological Polymers: Confined, Bent, and Driven, Ph.D. thesis, Vrije Universiteit Amsterdam, Amsterdam, 2013.
2.
J. M. Ball, Function Spaces for Liquid Crystals, slideshow from the Winter School on Nonlinear Function Spaces in Mathematics and Physical Sciences, 2015, https://people.maths.ox.ac.uk/ball/Teaching/lyon2015.pdf.
3.
P. J. Barratt and B. R. Duffy, Weak-anchoring effects on a Freedericksz transition in an annulus, Liquid Crystals, 19 (1995), pp. 57--63, https://doi.org/10.1080/02678299508036720.
4.
P. J. Barratt and B. R. Duffy, Freedericksz transitions in nematic liquid crystals in annular geometries, J. Phys. D, 29 (1996), pp. 1551--1558, http://stacks.iop.org/0022-3727/29/i=6/a=021.
5.
P. J. Barratt and B. R. Duffy, The effect of splay-bend elasticity on Freedericksz transitions in an annulus, Liquid Crystals, 26 (1999), pp. 743--751, https://doi.org/10.1080/026782999204831.
6.
P. Bauman, J. Park, and D. Phillips, Analysis of nematic liquid crystals with disclination lines, Arch. Ration. Mech. Anal., 205 (2012), pp. 795--826.
7.
F. Bethuel, H. Brezis, B. D. Coleman, and F. Hélein, Bifurcation analysis of minimizing harmonic maps describing the equilibrium of nematic phases between cylinders, Arch. Rational Mech. Anal., 118 (1992), pp. 149--168, https://doi.org/10.1007/BF00375093.
8.
F. Bethuel, H. Brezis, and F. Hélein, Ginzburg-Landau Vortices, Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser Boston, 1994, http://books.google.co.uk/books?id=FgEaYZq-wK8C.
9.
G. Bevilacqua and G. Napoli, Periodic splay-twist Fréedericksz transition for nematics confined between two concentric cylinders, Phys. Rev. E, 81 (2010), 031707, https://doi.org/10.1103/PhysRevE.81.031707.
10.
P. Biscari and E. G. Virga, Local stability of biaxial nematic phases between two cylinders, Int. J. Nonlinear. Mech., 32 (1997), pp. 337--351, https://doi.org/10.1016/S0020-7462(97)81142-0.
11.
F. Bisi, E. C. Gartland, R. Rosso, and E. G. Virga, Order reconstruction in frustrated nematic twist cells, Phys. Rev. E, 68 (2003), 021707, https://doi.org/10.1103/PhysRevE.68.021707.
12.
G. Canevari, A. Majumdar, and A. Spicer, Order reconstruction for nematics on squares and hexagons: A Landau--de Gennes study, SIAM J. Appl. Math., 77 (2017), pp. 267--293, https://doi.org/10.1137/16M1087990.
13.
O. J. Dammone, Confinement of Colloidal Liquid Crystals, Ph.D. thesis, University College, University of Oxford, Oxford, UK, 2013.
14.
A. J. Davidson and N. J. Mottram, Conformal mapping techniques for the modelling of liquid crystal devices, European J. Appl. Math., 23 (2012), pp. 99--119.
15.
P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd ed., Internat. Ser. Monogr. Phys., Oxford University Press, 1998.
16.
G. Di Fratta, J. M. Robbins, V. Slastikov, and A. Zarnescu, Half-integer point defects in the Q-tensor theory of nematic liquid crystals, J. Nonlinear Sci., 26 (2016), pp. 121--140, https://doi.org/10.1007/s00332-015-9271-8.
17.
I. C. Gârlea, P. Mulder, J. Alvarado, O. Dammone, D. G. A. L. Aarts, M. P. Lettinga, G. H. Koenderink, and B. M. Mulder, Finite particle size drives defect-mediated domain structures in strongly confined colloidal liquid crystals, Nature Commun., 7 (2016), 12112, https://doi.org/10.1038/ncomms12112.
18.
D. Golovaty and L. Berlyand, On uniqueness of vector-valued minimizers of the Ginzburg-Landau functional in annular domains, Calc. Var. Partial Differential Equations, 14 (2002), pp. 213--232, https://doi.org/10.1007/s005260100102.
19.
D. Golovaty, J. A. Montero, and P. Sternberg, Dimension reduction for the Landau-de Gennes model in planar nematic thin films, J. Nonlinear Sci., 25 (2015), pp. 1431--1451, https://doi.org/10.1007/s00332-015-9264-7.
20.
R. Ignat, L. Nguyen, V. Slastikov, and A. Zarnescu, Instability of point defects in a two-dimensional nematic liquid crystal model, Ann. Inst. H. Poincaré Anal. Non Linéare, 33 (2016), pp. 1131--1152.
21.
R. Ignat, L. Nguyen, V. Slastikov, and A. Zarnescu, Stability of point defects of degree $\pm\frac{1}{2}$ in a two-dimensional nematic liquid crystal model, Calc. Var. Partial Differential Equations, 55 (2016), 119, https://doi.org/10.1007/s00526-016-1051-2.
22.
G. Kitavtsev, J. M. Robbins, V. Slastikov, and A. Zarnescu, Liquid crystal defects in the Landau--de Gennes theory in two dimensions---Beyond the one-constant approximation, Math. Models Methods Appl. Sci., 26 (2016), pp. 2769--2808, https://doi.org/10.1142/S0218202516500664.
23.
S. Kralj and A. Majumdar, Order reconstruction patterns in nematic liquid crystal wells, Proc. Roy. Soc. A, 470 (2014), 20140276, https://doi.org/10.1098/rspa.2014.0276.
24.
H. Kusumaatmaja and A. Majumdar, Free energy pathways of a multistable liquid crystal device, Soft Matter, 11 (2015), pp. 4809--4817.
25.
X. Lamy, Some properties of the nematic radial hedgehog in the Landau--de Gennes theory, J. Math. Anal. Appl., 397 (2013), pp. 586--594, https://doi.org/10.1016/j.jmaa.2012.08.011.
26.
X. Lamy, Bifurcation analysis in a frustrated nematic cell, J. Nonlinear Sci., 24 (2014), pp. 1197--1230, https://doi.org/10.1007/s00332-014-9216-7.
27.
A. H. Lewis, D. G. A. L. Aarts, P. D. Howell, and A. Majumdar, Nematic equilibria on a two-dimensional annulus, Stud. Appl. Math., 138 (2017), pp. 438--466, https://doi.org/10.1111/sapm.12161.
28.
C. Luo, A. Majumdar, and R. Erban, Multistability in planar liquid crystal wells, Phys. Rev. E, 85 (2012), 061702, https://doi.org/10.1103/PhysRevE.85.061702.
29.
A. Majumdar, Equilibrium order parameters of nematic liquid crystals in the Landau-de Gennes theory, European J. Appl. Math., 21 (2010), pp. 181--203, https://doi.org/10.1017/S0956792509990210.
30.
A. Majumdar, The Landau-de Gennes theory of nematic liquid crystals: Uniaxiality versus biaxiality, Commun. Pure Appl. Anal., 11 (2012), pp. 1303--1337.
31.
A. Majumdar, The radial-hedgehog solution in Landau-de Gennes' theory for nematic liquid crystals, European J. Appl. Math., 23 (2012), pp. 61--97, https://doi.org/10.1017/S0956792511000295.
32.
A. Majumdar and A. Zarnescu, Landau-de Gennes theory of nematic liquid crystals: The Oseen-Frank limit and beyond, Arch. Ration. Mech. Anal., 196 (2010), pp. 227--280.
33.
O. V. Manyuhina, K. B. Lawlor, M. C. Marchetti, and M. J. Bowick, Viral nematics in confined geometries, Soft Matter, 11 (2015), pp. 6099--6105.
34.
P. Mironescu, On the stability of radial solutions of the Ginzburg-Landau equation, J. Funct. Anal., 130 (1995), pp. 334--344, https://doi.org/10.1006/jfan.1995.1073.
35.
S. Mkaddem and E. C. Gartland, Jr., Fine structure of defects in radial nematic droplets, Phys. Rev. E, 62 (2000), 6694.
36.
N. Mottram and C. Newton, Introduction to Q-tensor Theory, technical report, University of Strathclyde, Glasgow, UK, 2004.
37.
P. Palffy-Muhoray, A. Sparavigna, and A. Strigazzi, Saddle-splay and mechanical instability in nematics confined to a cylindrical annular geometry, Liquid Crystals, 14 (1993), pp. 1143--1151, https://doi.org/10.1080/02678299308027822.
38.
T.-W. Pan, Existence and multiplicity of radial solutions describing the equilibrium of nematic liquid crystals on annular domains, J. Math. Anal. Appl., 245 (2000), pp. 266--281, https://doi.org/10.1006/jmaa.2000.6780.
39.
C. Tsakonas, A. J. Davidson, C. V. Brown, and N. J. Mottram, Multistable alignment states in nematic liquid crystal filled wells, Appl. Phys. Lett., 90 (2007), 111913, https://doi.org/10.1063/1.2713140.
40.
E. G. Virga, Variational Theories for Liquid Crystals, Vol. 8, CRC Press, 1995.
Information & Authors
Information
Published In
SIAM Journal on Applied Mathematics
Pages: 1851 - 1875
DOI: 10.1137/17M1124656
ISSN (online): 1095-712X
Copyright
© 2017, Society for Industrial and Applied Mathematics.
History
Submitted: 7 April 2017
Accepted: 15 August 2017
Published online: 2 November 2017
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King Abdullah University of Science and Technology https://doi.org/10.13039/501100004052 : KUK-C1-013-04
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/J001686/1, EP/J001686/2
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