A non-traditional view on the modeling of nematic disclination dynamics
Authors:
Chiqun Zhang, Xiaohan Zhang, Amit Acharya, Dmitry Golovaty and Noel Walkington
Journal:
Quart. Appl. Math. 75 (2017), 309-357
MSC (2010):
Primary 76A15
DOI:
https://doi.org/10.1090/qam/1441
Published electronically:
August 18, 2016
MathSciNet review:
3614500
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Abstract: Non-singular disclination dynamics in a uniaxial nematic liquid crystal is modeled within a mathematical framework where the kinematics is a direct extension of the classical way of identifying these line defects with singularities of a unit vector field representing the nematic director. It is well known that the universally accepted Oseen-Frank energy is infinite for configurations that contain disclination line defects. We devise a natural augmentation of the Oseen-Frank energy to account for physical situations where, under certain conditions, infinite director gradients have zero associated energy cost, as would be necessary for modeling half-integer strength disclinations within the framework of the director theory. Equilibria and dynamics (in the absence of flow) of line defects are studied within the proposed model. Using appropriate initial/boundary data, the gradient-flow dynamics of this energy leads to non-singular, line defect equilibrium solutions, including those of half-integer strength. However, we demonstrate that the gradient flow dynamics for this energy is not able to adequately describe defect evolution. Motivated by similarity with dislocation dynamics in solids, a novel 2D-model of disclination dynamics in nematics is proposed. The model is based on the extended Oseen-Frank energy and takes into account thermodynamics and the kinematics of conservation of defect topological charge. We validate this model through computations of disclination equilibria, annihilation, repulsion, and splitting. We show that the energy function we devise, suitably interpreted, can serve as well for the modeling of equilibria and dynamics of dislocation line defects in solids, making the conclusions of this paper relevant to mechanics of both solids and liquid crystals.
References
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- Dmitry Golovaty and José Alberto Montero, On minimizers of a Landau–de Gennes energy functional on planar domains, Arch. Ration. Mech. Anal. 213 (2014), no. 2, 447–490. MR 3211856, DOI https://doi.org/10.1007/s00205-014-0731-3
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- Paolo Biscari and Timothy J. Sluckin, Field-induced motion of nematic disclinations, SIAM J. Appl. Math. 65 (2005), no. 6, 2141–2157. MR 2177743, DOI https://doi.org/10.1137/040618898
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- Eugene C. Gartland Jr., André M. Sonnet, and Epifanio G. Virga, Elastic forces on nematic point defects, Contin. Mech. Thermodyn. 14 (2002), no. 3, 307–319. MR 1913139, DOI https://doi.org/10.1007/s00161-002-0099-8
- R. Hardt, D. Kinderlehrer, and Fang-Hua Lin, Stable defects of minimizers of constrained variational principles, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), no. 4, 297–322 (English, with French summary). MR 963102
- L. Berlyand, D. Cioranescu, and D. Golovaty, Homogenization of a Ginzburg-Landau model for a nematic liquid crystal with inclusions, J. Math. Pures Appl. (9) 84 (2005), no. 1, 97–136 (English, with English and French summaries). MR 2112873, DOI https://doi.org/10.1016/j.matpur.2004.09.013
- Fang-Hua Lin and Chun Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math. 48 (1995), no. 5, 501–537. MR 1329830, DOI https://doi.org/10.1002/cpa.3160480503
- Fang-Hua Lin and Chun Liu, Existence of solutions for the Ericksen-Leslie system, Arch. Ration. Mech. Anal. 154 (2000), no. 2, 135–156. MR 1784963, DOI https://doi.org/10.1007/s002050000102
- Noel J. Walkington, Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations, ESAIM Math. Model. Numer. Anal. 45 (2011), no. 3, 523–540. MR 2804649, DOI https://doi.org/10.1051/m2an/2010065
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- Xiaohan Zhang, Amit Acharya, Noel J. Walkington, and Jacobo Bielak, A single theory for some quasi-static, supersonic, atomic, and tectonic scale applications of dislocations, J. Mech. Phys. Solids 84 (2015), 145–195. MR 3413434, DOI https://doi.org/10.1016/j.jmps.2015.07.004
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- Roberto Alicandro, Lucia De Luca, Adriana Garroni, and Marcello Ponsiglione, Metastability and dynamics of discrete topological singularities in two dimensions: a $\Gamma $-convergence approach, Arch. Ration. Mech. Anal. 214 (2014), no. 1, 269–330. MR 3237887, DOI https://doi.org/10.1007/s00205-014-0757-6
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- Alan C. Newell, Pattern quarks and leptons, Appl. Anal. 91 (2012), no. 2, 213–223. MR 2876750, DOI https://doi.org/10.1080/00036811.2011.619983
- A. Das, A. Acharya, J. Zimmer, and K. Matthies, Can equations of equilibrium predict all physical equilibria? A case study from field dislocation mechanics, Mathematics and Mechanics of Solids, vol. 18, no. 8, pp. 803–822, 2013.
- V. Vitek, Intrinsic stacking faults in body-centred cubic crystals, Philosophical Magazine, vol. 18, no. 154, pp. 773–786, 1968.
References
- I. W. Stewart, The static and dynamic continuum theory of liquid crystals: a mathematical introduction, CRC Press, 2004.
- Amit Acharya and Kaushik Dayal, Continuum mechanics of line defects in liquid crystals and liquid crystal elastomers, Quart. Appl. Math. 72 (2014), no. 1, 33–64. MR 3185131, DOI https://doi.org/10.1090/S0033-569X-2013-01322-X
- Hossein Pourmatin, Amit Acharya, and Kaushik Dayal, A fundamental improvement to Ericksen-Leslie kinematics, Quart. Appl. Math. 73 (2015), no. 3, 435–466. MR 3400752
- M. Kléman, Defect densities in directional media, mainly liquid crystals, Philosophical Magazine, vol. 27, no. 5, pp. 1057–1072, 1973.
- P.-G. de Gennes and J. Prost, The physics of liquid crystals (International Series of Monographs on Physics), Oxford University Press, no. 0.10, 1995, pp. 0–20.
- André M. Sonnet and Epifanio G. Virga, Dissipative ordered fluids, Theories for liquid crystals, Springer, New York, 2012. MR 2893657
- N. J. Mottram and C. J. Newton, Introduction to Q-tensor theory, preprint, arXiv:1409.3542, 2014.
- N. Schopohl and T. Sluckin, Defect core structure in nematic liquid crystals, Physical Review Letters, vol. 59, no. 22, p. 2582, 1987.
- Patricia Bauman, Jinhae Park, and Daniel Phillips, Analysis of nematic liquid crystals with disclination lines, Arch. Ration. Mech. Anal. 205 (2012), no. 3, 795–826. MR 2960033, DOI https://doi.org/10.1007/s00205-012-0530-7
- M. Ravnik and S. Žumer, Landau–de Gennes modelling of nematic liquid crystal colloids, Liquid Crystals, vol. 36, no. 10-11, pp. 1201–1214, 2009.
- 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), no. 1, 121–140. MR 3441275, DOI https://doi.org/10.1007/s00332-015-9271-8
- S. Kralj, S. Žumer, and D. W. Allender, Nematic-isotropic phase transition in a liquid-crystal droplet, Physical Review A, vol. 43, no. 6, p. 2943, 1991.
- Craig S. MacDonald, John A. Mackenzie, Alison Ramage, and Christopher J. P. Newton, Robust adaptive computation of a one-dimensional $\mathbf {Q}$-tensor model of nematic liquid crystals, Comput. Math. Appl. 64 (2012), no. 11, 3627–3640. MR 2992539, DOI https://doi.org/10.1016/j.camwa.2012.10.003
- Radu Ignat, Luc Nguyen, Valeriy Slastikov, and Arghir Zarnescu, Uniqueness results for an ODE related to a generalized Ginzburg-Landau model for liquid crystals, SIAM J. Math. Anal. 46 (2014), no. 5, 3390–3425. MR 3265181, DOI https://doi.org/10.1137/130948598
- Radu Ignat, Luc Nguyen, Valeriy Slastikov, and Arghir Zarnescu, Stability of the melting hedgehog in the Landau–de Gennes theory of nematic liquid crystals, Arch. Ration. Mech. Anal. 215 (2015), no. 2, 633–673. MR 3294413, DOI https://doi.org/10.1007/s00205-014-0791-4
- Radu Ignat, Luc Nguyen, Valeriy Slastikov, and Arghir Zarnescu, Stability of the vortex defect in the Landau-de Gennes theory for nematic liquid crystals, C. R. Math. Acad. Sci. Paris 351 (2013), no. 13-14, 533–537 (English, with English and French summaries). MR 3095101, DOI https://doi.org/10.1016/j.crma.2013.07.012
- L. Nguyen and A. Zarnescu, Refined approximation for a class of Landau-de Gennes energy minimizers, preprint, arXiv:1006.5689, 2010.
- P. Cladis and M. Kleman, Non-singular disclinations of strength s=+1 in nematics, Journal de Physique, vol. 33, no. 5-6, pp. 591–598, 1972.
- 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), no. 2, 149–168. MR 1158933, DOI https://doi.org/10.1007/BF00375093
- Paolo Biscari and Epifanio G. Virga, Local stability of biaxial nematic phases between two cylinders, Internat. J. Non-Linear Mech. 32 (1997), no. 2, 337–351. MR 1433928, DOI https://doi.org/10.1016/S0020-7462%2897%2981142-0
- Giacomo Canevari, Biaxiality in the asymptotic analysis of a 2D Landau–de Gennes model for liquid crystals, ESAIM Control Optim. Calc. Var. 21 (2015), no. 1, 101–137. MR 3348417, DOI https://doi.org/10.1051/cocv/2014025
- Ibrahim Fatkullin and Valeriy Slastikov, Vortices in two-dimensional nematics, Commun. Math. Sci. 7 (2009), no. 4, 917–938. MR 2604622
- Dmitry Golovaty and José Alberto Montero, On minimizers of a Landau–de Gennes energy functional on planar domains, Arch. Ration. Mech. Anal. 213 (2014), no. 2, 447–490. MR 3211856, DOI https://doi.org/10.1007/s00205-014-0731-3
- Duvan Henao and Apala Majumdar, Symmetry of uniaxial global Landau-de Gennes minimizers in the theory of nematic liquid crystals, SIAM J. Math. Anal. 44 (2012), no. 5, 3217–3241. MR 3023409, DOI https://doi.org/10.1137/110856861
- S. Kralj, E. G. Virga, and S. Žumer, Biaxial torus around nematic point defects, Physical Review E, vol. 60, no. 2, p. 1858, 1999.
- S. Mkaddem and E. Gartland Jr, Fine structure of defects in radial nematic droplets, Physical Review E, vol. 62, no. 5, p. 6694, 2000.
- F. C. Frank, I. Liquid crystals. On the theory of liquid crystals, Discussions of the Faraday Society, vol. 25, pp. 19–28, 1958.
- Epifanio G. Virga, Variational theories for liquid crystals, Applied Mathematics and Mathematical Computation, vol. 8, Chapman & Hall, London, 1994. MR 1369095
- Paolo Biscari and Timothy J. Sluckin, Expulsion of disclinations in nematic liquid crystals, European J. Appl. Math. 14 (2003), no. 1, 39–59. MR 1970236, DOI https://doi.org/10.1017/S0956792502005016
- Paolo Biscari and Timothy J. Sluckin, Field-induced motion of nematic disclinations, SIAM J. Appl. Math. 65 (2005), no. 6, 2141–2157 (electronic). MR 2177743, DOI https://doi.org/10.1137/040618898
- André M. Sonnet and Epifanio G. Virga, Dynamics of nematic loop disclinations, Phys. Rev. E (3) 56 (1997), no. 6, 6834–6842. MR 1492404, DOI https://doi.org/10.1103/PhysRevE.56.6834
- Eugene C. Gartland Jr., André M. Sonnet, and Epifanio G. Virga, Elastic forces on nematic point defects, Contin. Mech. Thermodyn. 14 (2002), no. 3, 307–319. MR 1913139, DOI https://doi.org/10.1007/s00161-002-0099-8
- R. Hardt, D. Kinderlehrer, and Fang-Hua Lin, Stable defects of minimizers of constrained variational principles, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), no. 4, 297–322 (English, with French summary). MR 963102
- L. Berlyand, D. Cioranescu, and D. Golovaty, Homogenization of a Ginzburg-Landau model for a nematic liquid crystal with inclusions, J. Math. Pures Appl. (9) 84 (2005), no. 1, 97–136 (English, with English and French summaries). MR 2112873, DOI https://doi.org/10.1016/j.matpur.2004.09.013
- Fang-Hua Lin and Chun Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math. 48 (1995), no. 5, 501–537. MR 1329830, DOI https://doi.org/10.1002/cpa.3160480503
- Fang-Hua Lin and Chun Liu, Existence of solutions for the Ericksen-Leslie system, Arch. Ration. Mech. Anal. 154 (2000), no. 2, 135–156. MR 1784963, DOI https://doi.org/10.1007/s002050000102
- Noel J. Walkington, Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations, ESAIM Math. Model. Numer. Anal. 45 (2011), no. 3, 523–540. MR 2804649, DOI https://doi.org/10.1051/m2an/2010065
- J. M. Ball and S. Bedford, Discontinuous order parameters in liquid crystal theories, Molecular Crystals and Liquid Crystals, vol. 612, no. 1, pp. 467–489, 2015.
- E. C. Gartland Jr, Scalings and limits of the Landau-de Gennes model for liquid crystals: A comment on some recent analytical papers, preprint, arXiv:1512.08164, 2015.
- C. Oseen, The theory of liquid crystals, Transactions of the Faraday Society, vol. 29, no. 140, pp. 883–899, 1933.
- J. L. Ericksen, Remarks concerning forces on line defects, Theoretical, experimental, and numerical contributions to the mechanics of fluids and solids, Z. Angew. Math. Phys. 46 (1995), Special Issue, S247–S271. MR 1359323
- J. L. Ericksen, Liquid crystals with variable degree of orientation, Arch. Rational Mech. Anal. 113 (1990), no. 2, 97–120. MR 1079183, DOI https://doi.org/10.1007/BF00380413
- Amit Acharya and Xiaohan Zhang, From dislocation motion to an additive velocity gradient decomposition, and some simple models of dislocation dynamics, Chin. Ann. Math. Ser. B 36 (2015), no. 5, 645–658. MR 3377868, DOI https://doi.org/10.1007/s11401-015-0970-0
- Xiaohan Zhang, Amit Acharya, Noel J. Walkington, and Jacobo Bielak, A single theory for some quasi-static, supersonic, atomic, and tectonic scale applications of dislocations, J. Mech. Phys. Solids 84 (2015), 145–195. MR 3413434, DOI https://doi.org/10.1016/j.jmps.2015.07.004
- J. Eshelby, The force on a disclination in a liquid crystal, Philosophical Magazine A, vol. 42, no. 3, pp. 359–367, 1980.
- Robert V. Kohn, Energy-driven pattern formation, International Congress of Mathematicians. Vol. I, Eur. Math. Soc., Zürich, 2007, pp. 359–383. MR 2334197, DOI https://doi.org/10.4171/022-1/15
- Robert Leon Jerrard and Halil Mete Soner, Dynamics of Ginzburg-Landau vortices, Arch. Rational Mech. Anal. 142 (1998), no. 2, 99–125. MR 1629646, DOI https://doi.org/10.1007/s002050050085
- Roberto Alicandro, Lucia De Luca, Adriana Garroni, and Marcello Ponsiglione, Metastability and dynamics of discrete topological singularities in two dimensions: a $\Gamma$-convergence approach, Arch. Ration. Mech. Anal. 214 (2014), no. 1, 269–330. MR 3237887, DOI https://doi.org/10.1007/s00205-014-0757-6
- N. M. Ercolani and S. C. Venkataramani, A variational theory for point defects in patterns, J. Nonlinear Sci. 19 (2009), no. 3, 267–300. MR 2511257, DOI https://doi.org/10.1007/s00332-008-9035-9
- Alan C. Newell, Pattern quarks and leptons, Appl. Anal. 91 (2012), no. 2, 213–223. MR 2876750, DOI https://doi.org/10.1080/00036811.2011.619983
- A. Das, A. Acharya, J. Zimmer, and K. Matthies, Can equations of equilibrium predict all physical equilibria? A case study from field dislocation mechanics, Mathematics and Mechanics of Solids, vol. 18, no. 8, pp. 803–822, 2013.
- V. Vitek, Intrinsic stacking faults in body-centred cubic crystals, Philosophical Magazine, vol. 18, no. 154, pp. 773–786, 1968.
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Additional Information
Chiqun Zhang
Affiliation:
Department of Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Email:
chiqunz@andrew.cmu.edu
Xiaohan Zhang
Affiliation:
Department of Mechanical Engineering, Stanford University, Stanford, California, 94305
MR Author ID:
1120090
Email:
xzhang11@stanford.edu
Amit Acharya
Affiliation:
Department of Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Email:
acharyaamit@cmu.edu
Dmitry Golovaty
Affiliation:
Department of Mathematics, University of Akron, Akron, Ohio 44325
MR Author ID:
617890
Email:
dmitry@uakron.edu
Noel Walkington
Affiliation:
Department of Mathematics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Email:
noelw@math.cmu.edu
Received by editor(s):
March 11, 2016
Published electronically:
August 18, 2016
Additional Notes:
The print version of this article is in black and white. Color is available online.
Article copyright:
© Copyright 2016
Brown University