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Modelling and computation of liquid crystals

Published online by Cambridge University Press:  04 August 2021

Wei Wang ,
Lei Zhang  and
Pingwen Zhang
Wei Wang
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou310027, China E-mail: [email protected]
Lei Zhang
Affiliation:
Beijing International Center for Mathematical Research, Center for Quantitative Biology, Peking University, Beijing100871, China E-mail: [email protected]
Pingwen Zhang
Affiliation:
School of Mathematical Sciences, Peking University, Beijing100871, China E-mail: [email protected]
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Abstract

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Liquid crystals are a type of soft matter that is intermediate between crystalline solids and isotropic fluids. The study of liquid crystals has made tremendous progress over the past four decades, which is of great importance for fundamental scientific research and has widespread applications in industry. In this paper we review the mathematical models and their connections to liquid crystals, and survey the developments of numerical methods for finding rich configurations of liquid crystals.

Type
Research Article
Information
Acta Numerica , Volume 30 , May 2021 , pp. 765 - 851
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Abels, H., Dolzmann, G. and Liu, Y. (2014), Well-posedness of a fully coupled Navier– Stokes/Q-tensor system with inhomogeneous boundary data, SIAM J. Math. Anal. 46, 3050–3077.CrossRefGoogle Scholar
Adler, J. H., Atherton, T. J., Emerson, D. B. and MacLachlan, S. P. (2015), An energy-minimization finite-element approach for the Frank–Oseen model of nematic liquid crystals, SIAM J. Numer. Anal. 53, 2226–2254.CrossRefGoogle Scholar
Adler, J. H., Emerson, D. B., MacLachlan, S. P. and Manteuffel, T. A. (2016), Constrained optimization for liquid crystal equilibria, SIAM J. Sci. Comput. 38, B50–B76.CrossRefGoogle Scholar
Alouges, F. (1997), A new algorithm for computing liquid crystal stable configurations: The harmonic mapping case, SIAM J. Numer. Anal. 34, 1708–1726.CrossRefGoogle Scholar
Ambrosio, L. (1990), Existence of minimal energy configurations of nematic liquid crystals with variable degree of orientation, Manuscripta Math. 68, 215–228.CrossRefGoogle Scholar
Angelani, L., Di Leonardo, R., Ruocco, G., Scala, A. and Sciortino, F. (2000), Saddles in the energy landscape probed by supercooled liquids, Phys. Rev. Lett. 85, 5356–5359.CrossRefGoogle ScholarPubMed
Badia, S., Guillén-González, F. and Gutiérrez-Santacreu, J. V. (2011a), Finite element approximation of nematic liquid crystal flows using a saddle-point structure, J. Comput. Phys. 230, 1686–1706.CrossRefGoogle Scholar
Badia, S., Guillén-Gónzalez, F. and Gutiérrez-Santacreu, J. V. (2011b), An overview on numerical analyses of nematic liquid crystal flows, Arch. Comput. Methods Eng. 18, 285.CrossRefGoogle Scholar
Bajc, I., Hecht, F. and Žumer, S. (2016), A mesh adaptivity scheme on the Landau–de Gennes functional minimization case in 3D, and its driving efficiency, J. Comput. Phys. 321, 981–996.CrossRefGoogle Scholar
Ball, J. M. (2017), Liquid crystals and their defects, in Mathematical Thermodynamics of Complex Fluids, Vol. 2200 of Lecture Notes in Mathematics, Springer, pp. 1–46.Google Scholar
Ball, J. M. (2020), Axisymmetry of critical points for the Onsager functional. Available at arXiv:2008.04009.Google Scholar
Ball, J. M. and Majumdar, A. (2010), Nematic liquid crystals: From Maier–Saupe to a continuum theory, Molecular Cryst. Liquid Cryst. 525, 1–11.CrossRefGoogle Scholar
Ball, J. M. and Zarnescu, A. (2008), Orientable and non-orientable line field models for uniaxial nematic liquid crystals, Molecular Cryst. Liquid Cryst. 495, 221–573.CrossRefGoogle Scholar
Ball, J. M. and Zarnescu, A. (2011), Orientability and energy minimization in liquid crystal models, Arch. Ration. Mech. Anal. 202, 493–535.CrossRefGoogle Scholar
Bauman, P., Park, J. and Phillips, D. (2012), Analysis of nematic liquid crystals with disclination lines, Arch. Ration. Mech. Anal. 205, 795–826.CrossRefGoogle Scholar
Becker, R., Feng, X. and Prohl, A. (2008), Finite element approximations of the Ericksen– Leslie model for nematic liquid crystal flow, SIAM J. Numer. Anal. 46, 1704–1731.CrossRefGoogle Scholar
Beris, A. N. and Edwards, B. J. (1994), Thermodynamics of Flowing Systems: With Internal Microstructure, Oxford Engineering Science Series, The Clarendon Press, Oxford University Press.CrossRefGoogle Scholar
Bisht, K., Wang, Y., Banerjee, V. and Majumdar, A. (2020), Tailored morphologies in two-dimensional ferronematic wells, Phys. Rev. E 101, 022706.CrossRefGoogle ScholarPubMed
Borthagaray, J. P., Nochetto, R. H. and Walker, S. W. (2020), A structure-preserving FEM for the uniaxially constrained Q-tensor model of nematic liquid crystals, Numer. Math. 145, 837–881.CrossRefGoogle Scholar
Brada, Z., Kralj, S., Svetec, M. and Zumer, S. (2003), Annihilation of nematic point defects: Postcollision scenarios, Phys. Rev. E 67, 050702.CrossRefGoogle Scholar
Brezis, H., Coron, J.-M. and Lieb, E. H. (1986), Harmonic maps with defects, Commun. Math. Phys. 107, 649–705.CrossRefGoogle Scholar
Cai, Y. and Wang, W. (2020), Global well-posedness for the three dimensional simplified inertial Ericksen–Leslie systems near equilibrium, J. Funct. Anal. 279, 108521.CrossRefGoogle Scholar
Cai, Y., Shen, J. and Xu, X. (2017a), A stable scheme and its convergence analysis for a 2D dynamic Q-tensor model of nematic liquid crystals, Math. Models Methods Appl. Sci. 27, 1459–1488.CrossRefGoogle Scholar
Cai, Y., Zhang, P. and Shi, A.-C. (2017b), Liquid crystalline bilayers self-assembled from rod–coil diblock copolymers, Soft Matter 13, 4607–4615.CrossRefGoogle Scholar
Canevari, G. (2015), Biaxiality in the asymptotic analysis of a 2D Landau–de Gennes model for liquid crystals, ESAIM Control Optim. Calc. Var. 21, 101–137.CrossRefGoogle Scholar
Canevari, G. (2017), Line defects in the small elastic constant limit of a three-dimensional Landau–de Gennes model, Arch. Ration. Mech. Anal. 223, 591–676.CrossRefGoogle Scholar
Canevari, G., Harris, J., Majumdar, A. and Wang, Y. W. (2020), The well order reconstruction solution for three-dimensional wells, in the Landau–de Gennes theory, Int. J. Nonlinear Mech. 119, 103342.CrossRefGoogle Scholar
Canevari, G., Majumdar, A. and Spicer, A. (2017), Order reconstruction for nematics on squares and hexagons: A Landau–de Gennes study, SIAM J. Appl. Math. 77, 267–293.CrossRefGoogle Scholar
Cerjan, C. J. and Miller, W. H. (1981), On finding transition states, J. Chem. Phys. 75, 2800–2806.CrossRefGoogle Scholar
Chaubal, C. V. and Leal, L. G. (1998), A closure approximation for liquid-crystalline polymer models based on parametric density estimation, J. Rheology 42, 177–201.CrossRefGoogle Scholar
Chen, J., Zhang, P. and Zhang, Z. (2018), Local minimizer and De Giorgi’s type conjecture for the isotropic–nematic interface problem, Calc. Var. Partial Differential Equations 57, 1–19.CrossRefGoogle Scholar
Chen, R., Bao, W. and Zhang, H. (2016), The kinematic effects of the defects in liquid crystal dynamics, Commun. Comput. Phys. 20, 234–249.CrossRefGoogle Scholar
Chen, R., Yang, X. and Zhang, H. (2017), Second order, linear, and unconditionally energy stable schemes for a hydrodynamic model of smectic-A liquid crystals, SIAM J. Sci. Comput. 39, A2808–A2833.CrossRefGoogle Scholar
Chen, W., Li, C. and Wang, G. (2010), On the stationary solutions of the 2D Doi–Onsager model, Nonlinear Anal. 73, 2410–2425.CrossRefGoogle Scholar
Chen, X., Korblova, E., Dong, D., Wei, X., Shao, R., Radzihovsky, L., Glaser, M. A., Maclennan, J. E., Bedrov, D., Walba, D. M. and Clark, N. A. (2020), First-principles experimental demonstration of ferroelectricity in a thermotropic nematic liquid crystal: Polar domains and striking electro-optics, Proc. Nat. Acad. Sci. 117, 14021–14031.CrossRefGoogle Scholar
Cohen, R. and Taylor, M. (1990), Weak stability of the map x/| x| for liquid crystal func-tionals, Commun. Partial Differential Equations 15, 675–692.CrossRefGoogle ScholarPubMed
Cohen, R., Hardt, R., Kinderlehrer, D., Lin, S.-Y. and Luskin, M. (1987), Minimum energy configurations for liquid crystals: Computational results, in Theory and Applications of Liquid Crystals, Vol. 5 of The IMA Volumes in Mathematics and Its Applications, Springer, pp. 99–121.CrossRefGoogle Scholar
Crippen, G. M. and Scheraga, H. A. (1971), Minimization of polypeptide energy XI: The method of gentlest ascent, Arch. Biochem. Biophys. 144, 462–466.CrossRefGoogle ScholarPubMed
Darmon, A., Benzaquen, M., Čopar, S., Dauchot, O. and Lopez-Leon, T. (2016), Topological defects in cholesteric liquid crystal shells, Soft Matter 12, 9280–9288.CrossRefGoogle ScholarPubMed
Davis, T. A. and Gartland, E. C. Jr (1998), Finite element analysis of the Landau–de Gennes minimization problem for liquid crystals, SIAM J. Numer. Anal. 35, 336–362.CrossRefGoogle Scholar
de Gennes, P. G. and Prost, J. (1993), The Physics of Liquid Crystals, Vol. 83 of International Series of Monographs on Physics, Oxford University Press.Google Scholar
de Luca, G. and Rey, A. D. (2007), Point and ring defects in nematics under capillary confinement, J. Chem. Phys. 127, 104902.CrossRefGoogle ScholarPubMed
Denniston, C., Orlandini, E. and Yeomans, J. M. (2001), Lattice Boltzmann simulations of liquid crystal hydrodynamics, Phys. Rev. E 63, 056702.CrossRefGoogle ScholarPubMed
Doi, M. (1981), Molecular dynamics and rheological properties of concentrated solutions of rodlike polymers in isotropic and liquid crystalline phases, J. Polymer Sci. Polymer Phys. Edn 19, 229–243.CrossRefGoogle Scholar
Doi, M. and Edwards, S. F. (1988), The Theory of Polymer Dynamics, Vol. 73 of International Series of Monographs on Physics, Oxford University Press, Oxford.Google Scholar
Doi, M., Zhou, J., Di, Y. and Xu, X. (2019), Application of the Onsager–Machlup integral in solving dynamic equations in nonequilibrium systems, Phys. Rev. E 99, 063303.CrossRefGoogle ScholarPubMed
Doye, J. P. K. and Wales, D. J. (2002), Saddle points and dynamics of Lennard-Jones clusters, solids, and supercooled liquids, J. Chem. Phys. 116, 3777–3788.CrossRefGoogle Scholar
Du, Q. and Feng, X. (2020), The phase field method for geometric moving interfaces and their numerical approximations, in Geometric Partial Differential Equations, Part I, Vol. 21 of Handbook of Numerical Analysis, Elsevier, pp. 425–508.CrossRefGoogle Scholar
Du, Q. and Zhang, L. (2009), A constrained string method and its numerical analysis, Commun. Math. Sci. 7, 1039–1051.CrossRefGoogle Scholar
Du, Q., Guo, B. and Shen, J. (2001), Fourier spectral approximation to a dissipative system modeling the flow of liquid crystals, SIAM J. Numer. Anal. 39, 735–762.CrossRefGoogle Scholar
E, W. (1997), Nonlinear continuum theory of smectic-A liquid crystals, Arch. Ration. Mech. Anal. 137, 159–175.Google Scholar
E, W. and Zhang, P. (2006), A molecular kinetic theory of inhomogeneous liquid crystal flow and the small Deborah number limit, Methods Appl. Anal. 13, 181–198.CrossRefGoogle Scholar
W, E and Zhou, X. (2011), The gentlest ascent dynamics, Nonlinearity 24, 1831.Google Scholar
W, E, Ren, W. and Vanden-Eijnden, E. (2002), String method for the study of rare events, Phys. Rev. B 66, 052301.Google Scholar
W, E, Ren, W. and Vanden-Eijnden, E. (2007), Simplified and improved string method for computing the minimum energy paths in barrier-crossing events, J. Chem. Phys. 126, 164103.Google Scholar
Elliott, C. M. and Stuart, A. M. (1993), The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal. 30, 1622–1663.CrossRefGoogle Scholar
Ericksen, J. L. (1961), Conservation laws for liquid crystals, Trans. Soc. Rheology 5, 23–34.CrossRefGoogle Scholar
Ericksen, J. L. (1990), Liquid crystals with variable degree of orientation, Arch. Ration. Mech. Anal. 113, 97–120.CrossRefGoogle Scholar
Eyre, D. J. (1998), Unconditionally gradient stable time marching the Cahn–Hilliard equation, in Symposia BB: Computational & Mathematical Models of Microstructural Evolution, Vol. 529 of Materials Research Society Symposium Proceedings, Materials Research Society, pp. 39–46.CrossRefGoogle Scholar
Farrell, P. E., Birkisson, Á. and Funke, S. W. (2015), Deflation techniques for finding distinct solutions of nonlinear partial differential equations, SIAM J. Sci. Statist. Comput. 37, A2026–A2045.CrossRefGoogle Scholar
Fatkullin, I. and Slastikov, V. (2005), Critical points of the Onsager functional on a sphere, Nonlinearity 18, 2565.CrossRefGoogle Scholar
Feng, J., Chaubal, C. and Leal, L. G. (1998), Closure approximations for the Doi theory: Which to use in simulating complex flows of liquid-crystalline polymers?, J. Rheology 42, 1095–1119.CrossRefGoogle Scholar
Feng, J. J., Sgalari, G. and Leal, L. G. (2000), A theory for flowing nematic polymers with orientational distortion, J. Rheology 44, 1085–1101.CrossRefGoogle Scholar
Feng, X. and Prohl, A. (2004), Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits, Math. Comp. 73, 541–567.CrossRefGoogle Scholar
Foffano, G., Lintuvuori, J. S., Tiribocchi, A. and Marenduzzo, D. (2014), The dynamics of colloidal intrusions in liquid crystals: A simulation perspective, Liquid Cryst. Rev. 2, 1–27.CrossRefGoogle Scholar
Frank, F. C. (1958), I. Liquid crystals. On the theory of liquid crystals, Discussions Faraday Soc. 25, 19–28.CrossRefGoogle Scholar
Friedel, G. (1922), Les états mésomorphes de la matière, Ann. de Physique 18, 273.CrossRefGoogle Scholar
Fukuda, J.-i, Stark, H., Yoneya, M. and Yokoyama, H. (2004), Interaction between two spherical particles in a nematic liquid crystal, Phys. Rev. E 69, 041706.CrossRefGoogle Scholar
Fukuda, J.-i., Žumer, S. et al. (2010), Novel defect structures in a strongly confined liquid-crystalline blue phase, Phys. Rev. Lett. 104, 017801.CrossRefGoogle Scholar
Gartland, E. C. Jr and Mkaddem, S. (1999), Instability of radial hedgehog configurations in nematic liquid crystals under Landau–de Gennes free-energy models, Phys. Rev. E 59, 563.CrossRefGoogle Scholar
Gartland, E. C. Jr Palffy-Muhoray, P. and Varga, R. S. (1991), Numerical minimization of the Landau–de Gennes free energy: Defects in cylindrical capillaries, Molecular Cryst. Liquid Cryst. 199, 429–452.CrossRefGoogle Scholar
Gelbart, W. M. and Ben-Shaul, A. (1982), Molecular theory of curvature elasticity in nematic liquids, J. Chem. Phys. 77, 916–933.CrossRefGoogle Scholar
Girault, V. and Guillén-González, F. (2011), Mixed formulation, approximation and de-coupling algorithm for a penalized nematic liquid crystals model, Math. Comp. 80, 781–819.CrossRefGoogle Scholar
Golovaty, D. and Montero, J. A. (2014), On minimizers of a Landau–de Gennes energy functional on planar domains, Arch. Ration. Mech. Anal. 213, 447–490.CrossRefGoogle Scholar
Gottarelli, G., Hibert, M., Samori, B., Solladie, G., Spada, G. P. and Zimmermann, R. (1983), Induction of the cholesteric mesophase in nematic liquid crystals: Mechanism and application to the determination of bridged biaryl configurations, J. Amer. Chem. Soc. 105, 7318–7321.CrossRefGoogle Scholar
Guillén-González, F. and Koko, J. (2015), A splitting in time scheme and augmented Lagrangian method for a nematic liquid crystal problem, J. Sci. Comput. 65, 1129–1144.CrossRefGoogle Scholar
Guillén-González, F. M. and Gutiérrez-Santacreu, J. V. (2013), A linear mixed finite element scheme for a nematic Ericksen–Leslie liquid crystal model, ESAIM Math. Model. Numer. Anal. 47, 1433–1464.CrossRefGoogle Scholar
Guo, Y., Afghah, S., Xiang, J., Lavrentovich, O. D., Selinger, R. L. and Wei, Q.-H. (2016), Cholesteric liquid crystals in rectangular microchannels: Skyrmions and stripes, Soft Matter 12, 6312–6320.CrossRefGoogle ScholarPubMed
Gupta, J. K., Sivakumar, S., Caruso, F. and Abbott, N. L. (2009), Size-dependent ordering of liquid crystals observed in polymeric capsules with micrometer and smaller diameters, Angewandte Chemie International Edition 48, 1652–1655.CrossRefGoogle ScholarPubMed
Han, J., Luo, Y., Wang, W., Zhang, P. and Zhang, Z. (2015), From microscopic theory to macroscopic theory: A systematic study on modeling for liquid crystals, Arch. Ration. Mech. Anal. 215, 741–809.CrossRefGoogle Scholar
Han, Y., Hu, Y., Zhang, P. and Zhang, L. (2019), Transition pathways between defect patterns in confined nematic liquid crystals, J. Comput. Phys. 396, 1–11.CrossRefGoogle Scholar
Han, Y., Majumdar, A. and Zhang, L. (2020a), A reduced study for nematic equilibria on two-dimensional polygons, SIAM J. Appl. Math. 80, 1678–1703.CrossRefGoogle Scholar
Han, Y., Yin, J., Zhang, P., Majumdar, A. and Zhang, L. (2020b), Solution landscape of a reduced Landau–de Gennes model on a hexagon. Available at arXiv:2003.07643.CrossRefGoogle Scholar
Hänggi, P., Talkner, P. and Borkovec, M. (1990), Reaction-rate theory: Fifty years after Kramers, Rev. Modern Phys. 62, 251.CrossRefGoogle Scholar
Hao, W. R., Hauenstein, J. D., Hu, B. and Sommese, A. J. (2014), A bootstrapping approach for computing multiple solutions of differential equations, J. Comput. Appl. Math. 258, 181–190.CrossRefGoogle Scholar
Hardt, R., Kinderlehrer, D. and Lin, F. (1986), Existence and partial regularity of static liquid crystal configurations, Commun. Math. Phys. 105, 547–570.CrossRefGoogle Scholar
Hélein, F. (1987), Minima de la fonctionnelle energie libre des cristaux liquides, CR Acad. Sci. Paris 305, 565–568.Google Scholar
Henkelman, G. and Jónsson, H. (1999), A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives, J. Chem. Phys. 111, 7010–7022.CrossRefGoogle Scholar
Hess, S. (1976), Fokker–Planck-equation approach to flow alignment in liquid crystals, Z. Naturforschung A 31, 1034–1037.CrossRefGoogle Scholar
Hieber, M. and Prüss, J. (2016), Modeling and analysis of the Ericksen–Leslie equations for nematic liquid crystal flows, in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids (Giga, Y. and Novotný, A., eds), Springer, pp. 1075–1134.Google Scholar
Hieber, M., Nesensohn, M., Prüss, J. and Schade, K. (2016), Dynamics of nematic liquid crystal flows: The quasilinear approach, Ann. Inst. H. Poincaré Anal. Non Linéaire 33, 397–408.CrossRefGoogle Scholar
Hinch, E. J. and Leal, L. G. (1976), Constitutive equations in suspension mechanics, part 2: Approximate forms for a suspension of rigid particles affected by Brownian rotations, J. Fluid Mech. 76, 187–208.CrossRefGoogle Scholar
Hong, M.-C. (2011), Global existence of solutions of the simplified Ericksen–Leslie system in dimension two, Calc. Var. Partial Differential Equations 40, 15–36.CrossRefGoogle Scholar
Hong, M.-C. and Xin, Z. (2012), Global existence of solutions of the liquid crystal flow for the Oseen–Frank model in <mono>R</mono>2, Adv. Math. 231, 1364–1400.CrossRefGoogle Scholar
Hu, Y., Qu, Y. and Zhang, P. (2016), On the disclination lines of nematic liquid crystals, Commun. Comput. Phys. 19, 354–379.CrossRefGoogle Scholar
Huang, J. and Ding, S. (2015), Global well-posedness for the dynamical Q-tensor model of liquid crystals, Sci. China Math. 58, 1349–1366.CrossRefGoogle Scholar
Huang, J., Lin, F. and Wang, C. (2014), Regularity and existence of global solutions to the Ericksen–Leslie system in <mono>R</mono>2 , Commun. Math. Phys. 331, 805–850.CrossRefGoogle Scholar
Huang, T., Lin, F., Liu, C. and Wang, C. (2016), Finite time singularity of the nematic liquid crystal flow in dimension three, Arch. Ration. Mech. Anal. 221, 1223–1254.CrossRefGoogle Scholar
Ignat, R., Nguyen, L., Slastikov, V. and Zarnescu, A. (2015), Stability of the melting hedgehog in the Landau–de Gennes theory of nematic liquid crystals, Arch. Ration. Mech. Anal. 215, 633–673.CrossRefGoogle Scholar
Ji, G., Yu, H. and Zhang, P. (2008), A kinetic-hydrodynamic simulation of liquid crystalline polymers under plane shear flow: 1 + 2 dimensional case, Commun. Comput. Phys. 4, 1194–1215.Google Scholar
Jiang, N. and Luo, Y.-L. (2019), On well-posedness of Ericksen–Leslie’s hyperbolic incom-pressible liquid crystal model, SIAM J. Math. Anal. 51, 403–434.CrossRefGoogle Scholar
Jónsson, H., Mills, G. and Jacobsen, K. W. (1998), Nudged elastic band method for finding minimum energy paths of transitions, in Classical and Quantum Dynamics in Condensed Phase Simulations (Berne, B. J., Ciccotti, G. and Coker, D. F., eds), World Scientific, pp. 385–404.CrossRefGoogle Scholar
Keber, F. C., Loiseau, E., Sanchez, T., DeCamp, S. J., Giomi, L., Bowick, M. J., Marchetti, M. C., Dogic, Z. and Bausch, A. R. (2014), Topology and dynamics of active nematic vesicles, Science 345, 1135–1139.CrossRefGoogle ScholarPubMed
Kinderlehrer, D., Walkington, N. and Ou, B. (1993), The elementary defects of the Oseen– Frank energy for a liquid crystal. Research report no. 93-NA-002, Carnegie Mellon University.Google Scholar
Kléman, M. (1989), Defects in liquid crystals, Rep. Progr. Phys. 52, 555.CrossRefGoogle Scholar
Knyazev, A. V. (2001), Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method, SIAM J. Sci. Comput. 23, 517–541.CrossRefGoogle Scholar
Kralj, S. and Majumdar, A. (2014), Order reconstruction patterns in nematic liquid crystal wells, Proc. Roy. Soc. A 470, 20140276.Google Scholar
Kralj, S., Virga, E. G. and Žumer, S. (1999), Biaxial torus around nematic point defects, Phys. Rev. E 60, 1858.CrossRefGoogle ScholarPubMed
Kusumaatmaja, H. and Majumdar, A. (2015), Free energy pathways of a multistable liquid crystal device, Soft Matter 11, 4809–4817.CrossRefGoogle ScholarPubMed
Kuzuu, N. and , M. Doi (1983), Constitutive equation for nematic liquid crystals under weak velocity gradient derived from a molecular kinetic equation, J. Phys. Soc. Japan 52, 3486–3494.CrossRefGoogle Scholar
Lai, C.-C., Lin, F., Wang, C., Wei, J. and Zhou, Y. (2019), Finite time blow-up for the nematic liquid crystal flow in dimension two. Available at arXiv:1908.10955.Google Scholar
Lamy, X. (2013), Some properties of the nematic radial hedgehog in the Landau–de Gennes theory, J. Math. Anal. Appl. 397, 586–594.CrossRefGoogle Scholar
Lamy, X. (2015), Uniaxial symmetry in nematic liquid crystals, Ann. Inst. H. Poincaré Anal. Non Linéaire 32, 1125–1144.CrossRefGoogle Scholar
Leslie, F. M. (1968), Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal. 28, 265–283.CrossRefGoogle Scholar
Leslie, F. M. (1979), Theory of flow phenomena in liquid crystals, Adv. Liquid Cryst. 4, 1–81.CrossRefGoogle Scholar
Li, S., Wang, W. and Zhang, P. (2015), Local well-posedness and small Deborah limit of a molecule-based Q-tensor system, Discrete Contin. Dyn. Syst. Ser. B 20, 2611–2655.CrossRefGoogle Scholar
Li, T., Zhang, P., Zhou, X. et al. (2004), Analysis of 1 + 1 dimensional stochastic models of liquid crystal polymer flows, Commun. Math. Sci. 2, 295–316.CrossRefGoogle Scholar
Li, Y. X. and Zhou, J. X. (2001), A minimax method for finding multiple critical points and its applications to semilinear PDEs, SIAM J. Sci. Statist. Comput. 23, 840–865.CrossRefGoogle Scholar
Lin, F. (1989), Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena, Commun. Pure Appl. Math. 42, 789–814.CrossRefGoogle Scholar
Lin, F. (1991), On nematic liquid crystals with variable degree of orientation, Comm. Pure Appl. Math. 44, 453–468.CrossRefGoogle Scholar
Lin, F. and Liu, C. (1995), Nonparabolic dissipative systems modeling the flow of liquid crystals, Commun. Pure Appl. Math. 48, 501–537.CrossRefGoogle Scholar
Lin, F. and Liu, C. (1996), Partial regularity of the dynamic system modeling the flow of liquid crystals, Discrete Contin. Dyn. Syst. A 2, 1–22.CrossRefGoogle Scholar
Lin, F. and Liu, C. (2001), Static and dynamic theories of liquid crystals, J. Partial Differ. Equ. 14, 289–330.Google Scholar
Lin, F. and Poon, C. C. (1996), On nematic liquid crystal droplets, in Elliptic and Parabolic Methods in Geometry (Minneapolis, MN, 1994), AK Peters, pp. 91–121.Google Scholar
Lin, F. and Wang, C. (2014), Recent developments of analysis for hydrodynamic flow of nematic liquid crystals, Philos. Trans. Roy. Soc. London A 372 (2029), 20130361.Google ScholarPubMed
Lin, F. and Wang, C. (2016), Global existence of weak solutions of the nematic liquid crystal flow in dimension three, Commun. Pure Appl. Math. 69, 1532–1571.CrossRefGoogle Scholar
Lin, F., Lin, J. and Wang, C. (2010), Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal. 197, 297–336.CrossRefGoogle Scholar
Lin, F., Pan, X. B. and Wang, C. (2012), Phase transition for potentials of high-dimensional wells, Commun. Pure Appl. Math. 65, 833–888.CrossRefGoogle Scholar
Lin, P. and Liu, C. (2006), Simulations of singularity dynamics in liquid crystal flows: A C 0 finite element approach, J. Comput. Phys. 215, 348–362.CrossRefGoogle Scholar
Lin, P., Liu, C. and Zhang, H. (2007), An energy law preserving C 0 finite element scheme for simulating the kinematic effects in liquid crystal dynamics, J. Comput. Phys. 227, 1411–1427.CrossRefGoogle Scholar
Liu, C. and Walkington, N. J. (2000), Approximation of liquid crystal flows, SIAM J. Numer. Anal. 37, 725–741.CrossRefGoogle Scholar
Liu, C. and Walkington, N. J. (2002), Mixed methods for the approximation of liquid crystal flows, ESAIM Math. Model. Numer. Anal. 36, 205–222.CrossRefGoogle Scholar
Liu, C. and Wang, Y. (2020a), On Lagrangian schemes for porous medium type generalized diffusion equations: A discrete energetic variational approach, J. Comput. Phys. 417, 109566.CrossRefGoogle Scholar
Liu, C. and Wang, Y. (2020b), A variational Lagrangian scheme for a phase field model: A discrete energetic variational approach. Available at arXiv:2003.10413.CrossRefGoogle Scholar
Liu, C., Shen, J. and Yang, X. (2007), Dynamics of defect motion in nematic liquid crystal flow: Modeling and numerical simulation, Commun. Comput. Phys 2, 1184–1198.Google Scholar
Liu, H., Zhang, H. and Zhang, P. (2005), Axial symmetry and classification of stationary solutions of Doi–Onsager equation on the sphere with Maier–Saupe potential, Commun. Math. Sci. 3, 201–218.CrossRefGoogle Scholar
Liu, Y. and Wang, W. (2018a), On the initial boundary value problem of a Navier–Stokes/Q-tensor model for liquid crystals, Discrete Contin. Dyn. Syst. B 23, 3879.Google Scholar
Liu, Y. and Wang, W. (2018b), The Oseen–Frank limit of Onsager’s molecular theory for liquid crystals, Arch. Ration. Mech. Anal. 227, 1061–1090.CrossRefGoogle Scholar
Liu, Y. and Wang, W. (2018c), The small Deborah number limit of the Doi–Onsager equation without hydrodynamics, J. Funct. Anal. 275, 2740–2793.CrossRefGoogle Scholar
Longa, L., Monselesan, D. and Trebin, H.-R. (1987), An extension of the Landau–Ginzburg– de Gennes theory for liquid crystals, Liquid Cryst. 2, 769–796.CrossRefGoogle Scholar
Lubensky, T. C., Pettey, D., Currier, N. and Stark, H. (1998), Topological defects and interactions in nematic emulsions, Phys. Rev. E 57, 610.CrossRefGoogle Scholar
MacDonald, C. S., Mackenzie, J. A. and Ramage, A. (2020), A moving mesh method for modelling defects in nematic liquid crystals, J. Comput. Phys. X 8, 100065.Google Scholar
Mahajan, V. N. and Dai, G. M. (2007), Orthonormal polynomials in wavefront analysis: Analytical solution, J. Optical Soc. Amer. A 24, 2994–3016.CrossRefGoogle ScholarPubMed
Maier, W. and Saupe, A. (1958), Eine einfache molekulare Theorie des nematischen kristallinflüssigen Zustandes, Z. Naturforschung A 13, 564–566.CrossRefGoogle Scholar
Majumdar, A. (2012), The radial-hedgehog solution in Landau–de Gennes’ theory for nematic liquid crystals, Europ. J. Appl. Math. 23, 61–97.CrossRefGoogle Scholar
Majumdar, A. and Wang, Y. (2018), Remarks on uniaxial solutions in the Landau–de Gennes theory, J. Math. Anal. Appl. 464, 328–353.CrossRefGoogle Scholar
Majumdar, A. and Zarnescu, A. (2010), Landau–de Gennes theory of nematic liquid crystals: The Oseen–Frank limit and beyond, Arch. Ration. Mech. Anal. 196, 227–280.CrossRefGoogle Scholar
Majumdar, A., Milewski, P. A. and Spicer, A. (2016), Front propagation at the nematic– isotropic transition temperature, SIAM J. Appl. Math. 76, 1296–1320.CrossRefGoogle Scholar
Majumdar, A., Newton, C. J. P., Robbins, J. M. and Zyskin, M. (2007), Topology and bistability in liquid crystal devices, Phys. Rev. E 75, 051703.CrossRefGoogle ScholarPubMed
Marchetti, M. C., Joanny, J.-F., Ramaswamy, S., Liverpool, T. B., Prost, J., Rao, M. and Simha, R. A. (2013), Hydrodynamics of soft active matter, Rev. Modern Phys. 85, 1143.CrossRefGoogle Scholar
Marrucci, G. and Greco, F. (1991), The elastic constants of Maier–Saupe rodlike molecule nematics, Molecular Cryst. Liquid Cryst. 206, 17–30.CrossRefGoogle Scholar
Mehta, D. (2011), Finding all the stationary points of a potential-energy landscape via numerical polynomial-homotopy-continuation method, Phys. Rev. E 84, 025702.CrossRefGoogle ScholarPubMed
Mertelj, A. and Lisjak, D. (2017), Ferromagnetic nematic liquid crystals, Liquid Cryst. Rev. 5, 1–33.CrossRefGoogle Scholar
Mertelj, A., Lisjak, D., Drofenik, M. and Čopič, M. (2013), Ferromagnetism in suspensions of magnetic platelets in liquid crystal, Nature 504, 237–241.CrossRefGoogle ScholarPubMed
Milnor, J. W., Spivak, M. and Wells, R. (1969), Morse Theory, Vol. 1, Princeton University Press.Google Scholar
Miron, R. A. and Fichthorn, K. A. (2001), The step and slide method for finding saddle points on multidimensional potential surfaces, J. Chem. Phys. 115, 8742–8747.CrossRefGoogle Scholar
Mkaddem, S. and Gartland, E. C. Jr (2000), Fine structure of defects in radial nematic droplets, Phys. Rev. E 62, 6694.CrossRefGoogle ScholarPubMed
Mottram, N. J. and Hogan, S. J. (1997), Disclination core structure and induced phase change in nematic liquid crystals, Philos. Trans. Roy. Soc. London A 355, 2045–2064.CrossRefGoogle Scholar
Mottram, N. J. and Sluckin, T. J. (2000), Defect-induced melting in nematic liquid crystals, Liquid Cryst. 27, 1301–1304.CrossRefGoogle Scholar
Mousseau, N. and Barkema, G. T. (1998), Traveling through potential energy landscapes of disordered materials: The activation–relaxation technique, Phys. Rev. E 57, 2419.CrossRefGoogle Scholar
Muller, M., Smirnova, Y. G., Marelli, G., Fuhrmans, M. and Shi, A.-C. (2012), Transition path from two apposed membranes to a stalk obtained by a combination of particle simulations and string method, Phys. Rev. Lett. 108, 228103.CrossRefGoogle ScholarPubMed
Muševič, I., Škarabot, M., Tkalec, U., Ravnik, M. and Žumer, S. (2006), Two-dimensional nematic colloidal crystals self-assembled by topological defects, Science 313 (5789), 954–958.CrossRefGoogle ScholarPubMed
Nestler, M., Nitschke, I., Praetorius, S. and Voigt, A. (2018), Orientational order on surfaces: The coupling of topology, geometry, and dynamics, J. Nonlinear Sci. 28, 147–191.CrossRefGoogle Scholar
Nguyen, L. and Zarnescu, A. (2013), Refined approximation for minimizers of a Landau–de Gennes energy functional, Calc. Var. Partial Differential Equations 47, 383–432.CrossRefGoogle Scholar
Nitschke, I., Nestler, M., Praetorius, S., Löwen, H. and Voigt, A. (2018), Nematic liquid crystals on curved surfaces: A thin film limit, Proc. Roy. Soc. A 474, 20170686.CrossRefGoogle ScholarPubMed
Nochetto, R. H., Walker, S. W. and Zhang, W. (2017), A finite element method for nematic liquid crystals with variable degree of orientation, SIAM J. Numer. Anal. 55, 1357–1386.CrossRefGoogle Scholar
Noh, J., Wang, Y., Liang, H.-L., Subba, V., Jampani, R., Majumdar, A. and Lager-wall, J. P. F. (2020), Dynamic tuning of the director field in liquid crystal shells using block copolymers, Phys. Rev. E 2, 033160.Google Scholar
Oh-e, M. and Kondo, K. (1995), Electro-optical characteristics and switching behavior of the in-plane switching mode, Appl. Phys. Lett. 67, 3895–3897.CrossRefGoogle Scholar
Onsager, L. (1949), The effects of shape on the interaction of colloidal particles, Ann. New York Acad. Sci. 51, 627–659.CrossRefGoogle Scholar
Oseen, C. W. (1933), The theory of liquid crystals, Trans. Faraday Soc. 29, 883–899.CrossRefGoogle Scholar
Paicu, M. and Zarnescu, A. (2012), Energy dissipation and regularity for a coupled Navier– Stokes and Q-tensor system, Arch. Ration. Mech. Anal. 203, 45–67.CrossRefGoogle Scholar
Park, J., Wang, W., Zhang, P. and Zhang, Z. (2017), On minimizers for the isotropic–nematic interface problem, Calc. Var. Partial Differential Equations 56, 41.CrossRefGoogle Scholar
Parodi, O. (1970), Stress tensor for a nematic liquid crystal, J. de Physique 31, 581–584.CrossRefGoogle Scholar
Prost, J., Jülicher, F. and Joanny, J.-F. (2015), Active gel physics, Nature Phys. 11, 111–117.CrossRefGoogle Scholar
Qian, T. and Sheng, P. (1998), Generalized hydrodynamic equations for nematic liquid crystals, Phys. Rev. E 58, 7475.CrossRefGoogle Scholar
Qu, Y., Wei, Y. and Zhang, P. (2017), Transition of defect patterns from 2D to 3D in liquid crystals, Commun. Comput. Phys. 21, 890–904.CrossRefGoogle Scholar
Ramage, A. and Sonnet, A. M. (2016), Computational fluid dynamics for nematic liquid crystals, BIT Numer. Math. 56, 573–586.CrossRefGoogle Scholar
Ravnik, M. and Žumer, S. (2009), Landau–de Gennes modelling of nematic liquid crystal colloids, Liquid Cryst. 36, 1201–1214.CrossRefGoogle Scholar
Ren, W. and Vanden-Eijnden, E. (2013), A climbing string method for saddle point search, J. Chem. Phys. 138, 134105.CrossRefGoogle ScholarPubMed
Rey, A. D. and Tsuji, T. (1998), Recent advances in theoretical liquid crystal rheology, Macromolecular Theory Simul. 7, 623–639.3.0.CO;2-E>CrossRefGoogle Scholar
Robinson, M., Luo, C., Farrell, P. E., Erban, R. and Majumdar, A. (2017), From molecular to continuum modelling of bistable liquid crystal devices, Liquid Cryst. 44, 2267–2284.CrossRefGoogle Scholar
Samanta, A. and W, E (2013), Optimization-based string method for finding minimum energy path, Commun. Comput. Phys. 14, 265–275.CrossRefGoogle Scholar
Shen, J. and Yang, X. (2010), Numerical approximations of Allen–Cahn and Cahn–Hilliard equations, Discrete Contin. Dyn. Syst. A 28, 1669.CrossRefGoogle Scholar
Shen, J., Xu, J. and Yang, J. (2018), The scalar auxiliary variable (SAV) approach for gradient flows, J. Comput. Phys. 353, 407–416.CrossRefGoogle Scholar
Shen, J., Xu, J. and Yang, J. (2019), A new class of efficient and robust energy stable schemes for gradient flows, SIAM Rev. 61, 474–506.CrossRefGoogle Scholar
Shen, Q., Liu, C. and Calderer, M. C. (2002), Axisymmetric configurations of bipolar liquid crystal droplets, Contin. Mech. Thermodyn. 14, 363–375.CrossRefGoogle Scholar
Siavashpouri, M., Wachauf, C. H., Zakhary, M. J., Praetorius, F., Dietz, H. and Dogic, Z. (2017), Molecular engineering of chiral colloidal liquid crystals using DNA origami, Nature Materials 16, 849–856.CrossRefGoogle ScholarPubMed
Sonnet, A., Kilian, A. and Hess, S. (1995), Alignment tensor versus director: Description of defects in nematic liquid crystals, Phys. Rev. E 52, 718.CrossRefGoogle ScholarPubMed
Stephen, K. and Adrian, G. (2002), Controllable alignment of nematic liquid crystals around microscopic posts: Stabilization of multiple states, Appl. Phys. Lett. 80, 3635–3637.Google Scholar
Stewart, I. W. (2004), The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction, Taylor & Francis.Google Scholar
Takashi, T. and Yasumasa, N. (2010), Morphological characterization of the diblock copolymer problem with topological computation, Japan. J. Indust. Appl. Math. 27, 175–190.Google Scholar
Taylor, J. M. (2018), Oseen–Frank-type theories of ordered media as the Γ-limit of a non-local mean-field free energy, Math. Models Methods Appl. Sci. 28, 615–657.CrossRefGoogle Scholar
Tong, Y., Wang, Y. and Zhang, P. (2017), Defects around a spherical particle in cholesteric liquid crystals, Numer. Math. Theory Methods Appl. 10, 205–221.CrossRefGoogle Scholar
Vanden-Eijnden, E. et al. (2010), Transition-path theory and path-finding algorithms for the study of rare events, Ann. Rev. Phys. Chem. 61, 391–420.Google Scholar
Vincent, T. L., Goh, B. S. and Teo, K. L. (1992), Trajectory-following algorithms for min-max optimization problems, J. Optim. Theory Appl. 75, 501–519.CrossRefGoogle Scholar
Vollmer, M. A. C. (2017), Critical points and bifurcations of the three-dimensional Onsager model for liquid crystals, Arch. Ration. Mech. Anal. 226, 851–922.CrossRefGoogle Scholar
Vukadinovic, J. (2009), Inertial manifolds for a Smoluchowski equation on the unit sphere, Commun. Math. Phys. 285, 975–990.CrossRefGoogle Scholar
Walker, S. W. (2020), A finite element method for the generalized Ericksen model of nematic liquid crystals, ESAIM Math. Model. Numer. Anal. 54, 1181–1220.CrossRefGoogle Scholar
Walkington, N. J. (2011), Numerical approximation of nematic liquid crystal flows governed by the Ericksen–Leslie equations, ESAIM Math. Model. Numer. Anal. 45, 523–540.CrossRefGoogle Scholar
Walters, M., Wei, Q. and Chen, J. Z. Y. (2019), Machine learning topological defects of confined liquid crystals in two dimensions, Phys. Rev. E 99, 062701.CrossRefGoogle ScholarPubMed
Wang, Q. (2002), A hydrodynamic theory for solutions of nonhomogeneous nematic liquid crystalline polymers of different configurations, J. Chem. Phys. 116, 9120–9136.CrossRefGoogle Scholar
Wang, Q., W, E, Liu, C. and Zhang, P. (2002), Kinetic theory for flows of nonhomogeneous rodlike liquid crystalline polymers with a nonlocal intermolecular potential, Phys. Rev. E 65, 051504.CrossRefGoogle ScholarPubMed
Wang, W., Zhang, P. and Zhang, Z. (2013), Well-posedness of the Ericksen–Leslie system, Arch. Ration. Mech. Anal. 210, 837–855.CrossRefGoogle Scholar
Wang, W., Zhang, P. and Zhang, Z. (2015a), Rigorous derivation from Landau–de Gennes theory to Ericksen–Leslie theory, SIAM J. Math. Anal. 47, 127–158.CrossRefGoogle Scholar
Wang, W., Zhang, P. and Zhang, Z. (2015b), The small Deborah number limit of the Doi– Onsager equation to the Ericksen–Leslie equation, Commun. Pure Appl. Math. 68, 1326–1398.CrossRefGoogle Scholar
Wang, Y., Canevari, G. and Majumdar, A. (2019), Order reconstruction for nematics on squares with isotropic inclusions: A Landau–de Gennes study, SIAM J. Appl. Math. 79, 1314–1340.CrossRefGoogle Scholar
Wang, Y., Zhang, P. and Chen, J. Z. Y. (2017), Topological defects in an unconfined nematic fluid induced by single and double spherical colloidal particles, Phys. Rev. E 96, 042702.CrossRefGoogle Scholar
Wang, Y., Zhang, P. and Chen, J. Z. Y. (2018), Formation of three-dimensional colloidal crystals in a nematic liquid crystal, Soft Matter 14, 6756–6766.CrossRefGoogle Scholar
Wilkinson, M. (2015), Strictly physical global weak solutions of a Navier–Stokes Q-tensor system with singular potential, Arch. Ration. Mech. Anal. 218, 487–526.CrossRefGoogle Scholar
Willman, E., Fernández, F. A., James, R. and Day, S. E. (2008), Switching dynamics of a post-aligned bistable nematic liquid crystal device, J. Display Technol. 4, 276–281.CrossRefGoogle Scholar
Xu, J. and Zhang, P. (2014), From microscopic theory to macroscopic theory: Symmetries and order parameters of rigid molecules, Sci. China Math. 57, 443–468.CrossRefGoogle Scholar
Xu, J. and Zhang, P. (2018), Calculating elastic constants of bent-core molecules from Onsager-theory-based tensor model, Liquid Cryst. 45, 22–31.CrossRefGoogle Scholar
Xu, J., Li, Y., Wu, S. and Bousquet, A. (2019), On the stability and accuracy of partially and fully implicit schemes for phase field modeling, Comput. Methods Appl. Mech. Engrg 345, 826–853.CrossRefGoogle Scholar
Xu, J., Ye, F. and Zhang, P. (2018), A tensor model for nematic phases of bent-core molecules based on molecular theory, Multiscale Model. Simul. 16, 1581–1602.CrossRefGoogle Scholar
Yang, X. (2016), Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends, J. Comput. Phys. 327, 294–316.CrossRefGoogle Scholar
Yin, J., Wang, Y., Chen, J. Z. Y., Zhang, P. and Zhang, L. (2020a), Construction of a pathway map on a complicated energy landscape, Phys. Rev. Lett. 124, 090601.CrossRefGoogle Scholar
Yin, J., Yu, B. and Zhang, L. (2020b), Searching the solution landscape by generalized high-index saddle dynamics, Sci. China Math. doi:10.1007/s11425-020-1737-1.CrossRefGoogle Scholar
Yin, J., Zhang, L. and Zhang, P. (2019), High-index optimization-based shrinking dimer method for finding high-index saddle points, SIAM J. Sci. Comput. 41, A3576–3595.CrossRefGoogle Scholar
Yu, H. and Zhang, P. (2007), A kinetic–hydrodynamic simulation of microstructure of liquid crystal polymers in plane shear flow, J. Non-Newtonian Fluid Mech. 141, 116–127.CrossRefGoogle Scholar
Yu, H., Ji, G. and Zhang, P. (2010), A nonhomogeneous kinetic model of liquid crystal polymers and its thermodynamic closure approximation, Commun. Comput. Phys. 7, 383–402.CrossRefGoogle Scholar
Yu, Y. (2020), Disclinations in limiting Landau–de Gennes theory, Arch. Ration. Mech. Anal. 237, 147–200.CrossRefGoogle Scholar
Yue, P., Feng, J. J., Liu, C. and Shen, J. (2004), A diffuse-interface method for simulating two-phase flows of complex fluids, J. Fluid Mech. 515, 293.CrossRefGoogle Scholar
Zernike, F. (1934), Diffraction theory of the knife-edge test and its improved form, the phase-contrast method, Month. Not. Roy. Astronom. Soc. 94, 377–384.CrossRefGoogle Scholar
Zhang, H. and Zhang, P. (2007), Stable dynamic states at the nematic liquid crystals in weak shear flow, Phys. D 232, 156–165.CrossRefGoogle Scholar
Zhang, H. and Zhang, P. (2008), On the new multiscale rodlike model of polymeric fluids, SIAM J. Math. Anal. 40, 1246–1271.CrossRefGoogle Scholar
Zhang, J. and Du, Q. (2012), Shrinking dimer dynamics and its applications to saddle point search, SIAM J. Numer. Anal. 50, 1899–1921.CrossRefGoogle Scholar
Zhang, L., Du, Q. and Zheng, Z. Z. (2016a), Optimization-based shrinking dimer method for finding transition states, SIAM J. Sci. Comput. 38, A528–A544.CrossRefGoogle Scholar
Zhang, L., Ren, W. Q., Samanta, A. and Du, Q. (2016b), Recent developments in computational modelling of nucleation in phase transformations, NPJ Comput. Materials 2, 16003.CrossRefGoogle Scholar
Zhao, J. and Wang, Q. (2016), Semi-discrete energy-stable schemes for a tensor-based hydrodynamic model of nematic liquid crystal flows, J. Sci. Comput. 68, 1241–1266.CrossRefGoogle Scholar
Zhao, J., Yang, X., Gong, Y. and Wang, Q. (2017), A novel linear second order unconditionally energy stable scheme for a hydrodynamic Q-tensor model of liquid crystals, Comput. Methods Appl. Mech. Engrg 318, 803–825.CrossRefGoogle Scholar
Zhao, J., Yang, X., Li, J. and Wang, Q. (2016), Energy stable numerical schemes for a hydrodynamic model of nematic liquid crystals, SIAM J. Sci. Comput. 38, A3264–A3290.CrossRefGoogle Scholar
Zhou, C., Yue, P. and Feng, J. J. (2007), The rise of Newtonian drops in a nematic liquid crystal, J. Fluid Mech. 593, 385–404.CrossRefGoogle Scholar
Zhou, H., Wang, H., Forest, M. G. and Wang, Q. (2005), A new proof on axisymmetric equilibria of a three-dimensional Smoluchowski equation, Nonlinearity 18, 2815–2825.CrossRefGoogle Scholar
Zhou, S., Sokolov, A., Lavrentovich, O. D. and Aranson, I. S. (2014), Living liquid crystals, Proc. Nat. Acad. Sci. 111, 1265–1270.CrossRefGoogle ScholarPubMed
Zhu, T., Li, J., Samanta, A., Kim, H. G. and Suresh, S. (2007), Interfacial plasticity governs strain rate sensitivity and ductility in nanostructured metals, Proc. Nat. Acad. Sci. 104, 3031–3036.CrossRefGoogle ScholarPubMed