# American Institute of Mathematical Sciences

April  2015, 8(2): 283-302. doi: 10.3934/dcdss.2015.8.283

## A Landau--de Gennes theory of liquid crystal elastomers

 1 School of Mathematics, University of Minnesota, 206 Church Street S.E, Minneapolis, MN 55455, United States, United States 2 Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824

Received  July 2013 Revised  November 2013 Published  July 2014

In this article, we study minimization of the energy of a Landau-de Gennes liquid crystal elastomer. The total energy consists of the sum of the Lagrangian elastic stored energy function of the elastomer and the Eulerian Landau-de Gennes energy of the liquid crystal.
There are two related sources of anisotropy in the model, that of the rigid units represented by the traceless nematic order tensor $Q$, and the positive definite step-length tensor $L$ characterizing the anisotropy of the network. This work is motivated by the study of cytoskeletal networks which can be regarded as consisting of rigid rod units crosslinked into a polymeric-type network. Due to the mixed Eulerian-Lagrangian structure of the energy, it is essential that the deformation maps $\varphi$ be invertible. For this, we require sufficient regularity of the fields $(\varphi, Q)$ of the problem, and that the deformation map satisfies the Ciarlet-Nečas injectivity condition. These, in turn, determine what boundary conditions are admissible, which include the case of Dirichlet conditions on both fields. Alternatively, the approach of including the Rapini-Papoular surface energy for the pull-back tensor $\tilde Q$ is also discussed. The regularity requirements also lead us to consider powers of the gradient of the order tensor $Q$ higher than quadratic in the energy.
We assume polyconvexity of the stored energy function with respect to the effective deformation tensor and apply methods of calculus of variations from isotropic nonlinear elasticity. Recovery of minimizing sequences of deformation gradients from the corresponding sequences of effective deformation tensors requires invertibility of the anisotropic shape tensor $L$. We formulate a necessary and sufficient condition to guarantee this invertibility property in terms of the growth to infinity of the bulk liquid crystal energy $f(Q)$, as the minimum eigenvalue of $Q$ approaches the singular limit of $-\frac{1}{3}$. It turns out that $L$ becomes singular as the minimum eigenvalue of $Q$ reaches $-\frac{1}{3}$. Lower bounds on the eigenvalues of $Q$ are needed to ensure compatibility between the theories of Landau-de Gennes and Maier-Saupe of nematics [5].
Citation: M. Carme Calderer, Carlos A. Garavito Garzón, Baisheng Yan. A Landau--de Gennes theory of liquid crystal elastomers. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 283-302. doi: 10.3934/dcdss.2015.8.283
##### References:
 [1] V. Agostiniani and A. DeSimone, Ogden-type energies for nematic elastomers,, Int J. Nonlinear Mechanics, 47 (2012), 402. Google Scholar [2] D. Anderson, D. Carlson and E. Fried, A continuum-mechanical theory for nematic elastomers,, J. Elasticity, 56 (1999), 33. doi: 10.1023/A:1007647913363. Google Scholar [3] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity,, Archive for Rational Mechanics and Analysis, 63 (1977), 337. doi: 10.1007/BF00279992. Google Scholar [4] J. M. Ball, Global invertibility of sobolev functions and the interpenetration of matter,, Proceedings Royal Soc. Edinburgh A, 88 (1981), 315. doi: 10.1017/S030821050002014X. Google Scholar [5] J. M. Ball and A. Majumdar, Nematic liquid crystals: From maier-saupe to a continuum theory,, Molecular Crystals and Liquid Crystals, 525 (2010), 1. doi: 10.1080/15421401003795555. Google Scholar [6] M. Barchiesi and A. DeSimone, Frank energy for nematic elastomers. A nonlinear model,, Preprint, (2013). Google Scholar [7] M. C. Calderer and B. Mukherjee, Chevron patterns in liquid crystal flows,, Physica D, 98 (1996), 201. doi: 10.1016/0167-2789(96)00051-6. Google Scholar [8] M. C. Calderer and B. Mukherjee, On poiseuille flow of liquid crystals,, Liquid Crystals, 22 (1997), 121. doi: 10.1080/026782997209487. Google Scholar [9] M. C. Calderer and B. Mukherjee, Mathematical issues in the modeling of flow behavior of polymeric liquid crystals,, J. Rheol., 42 (1998), 1519. Google Scholar [10] M. C. Calderer, Critical size of stripe patterns in liquid crystal elastomers under extension,, Preprint, (2013). Google Scholar [11] M. C. Calderer, C. A. Garavito and C. Luo, Liquid Crystal Elastomers and Phase Transitions in Rod Networks,, Technical Report, (2013). Google Scholar [12] M. C. Calderer and C. Liu, Liquid crystal flow: dynamic and static configurations,, SIAM Journal on Applied Mathematics, 20 (2000), 1225. Google Scholar [13] M. C. Calderer, C. Liu and B. Yan, A mathematical theory for nematic elastomers with non-uniform prolate spheroids,, in Advances in Applied and Computational Mathematics (eds. F. Liu, (2006), 245. Google Scholar [14] M. C. Calderer and C. Luo, Numerical study of liquid crystal elastomers by a mixed finite element method,, European J. Appl. Math., 23 (2012), 121. doi: 10.1017/S0956792511000313. Google Scholar [15] P. Cesana and A. DeSimone, Strain-order coupling in nematic elastomers: Equilibrium configurations,, Math. Models Methods Appl. Sci, 19 (2009), 601. doi: 10.1142/S0218202509003541. Google Scholar [16] P. Cesana and A. DeSimone, Quasiconvex envelopes of energies for nematic elastomers in the small strain regime and applications,, J. Mech. Phys. Solids, 59 (2011), 787. doi: 10.1016/j.jmps.2011.01.007. Google Scholar [17] P-G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity,, Arch. Rational Mech. Anal., 97 (1987), 171. doi: 10.1007/BF00250807. Google Scholar [18] S. Conti, A. DeSimone and G. Dolzmann, Soft elastic response of stretched sheets of nematic elastomers: A numerical study,, Journal of the Mechanics and Physics of Solids, 50 (2002), 1431. doi: 10.1016/S0022-5096(01)00120-X. Google Scholar [19] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals,, Oxford University Press, (1995). Google Scholar [20] A. DeSimone and G. Dolzmann, Material instabilities in nematic elastomers,, Physica D, 136 (2000), 175. doi: 10.1016/S0167-2789(99)00153-0. Google Scholar [21] A. DeSimone and G. Dolzmann, Macroscopic response of nematic elastomers via relaxation of a class of $SO(3)$-invariant energies,, Arch. Rat. Mech. Anal., 161 (2002), 181. doi: 10.1007/s002050100174. Google Scholar [22] A. DeSimone and L. Teresi, Elastic energies for nematic elastomers,, The European Physical Journal E: Soft Matter and Biological Physics, 29 (2009), 191. doi: 10.1140/epje/i2009-10467-9. Google Scholar [23] J. L. Ericksen, Liquid crystals with variable degree of orientation,, Archive for Rational Mechanics and Analysis, 113 (1991), 97. doi: 10.1007/BF00380413. Google Scholar [24] I. Fonseca and W. Gangbo, Local invertibility of sobolev functions,, SIAM J. Math. Anal., 26 (1995), 280. doi: 10.1137/S0036141093257416. Google Scholar [25] G. Forest, Q. Wang and R. Zhou, A kinetic theory for solutions of nonhomogeneous nematic liquid crystalline polymers with density variations,, J.Fluid Engineering, 126 (2004), 180. Google Scholar [26] E. Fried and S. Sellers, Free-energy density functions for nematic elastomers,, J. Mech. Phys. Solids, 52 (1999), 1671. doi: 10.1016/j.jmps.2003.12.005. Google Scholar [27] E. Fried and S. Sellers, Soft elasticity is not necessary for striping in nematic elastomers,, Journal of Applied Physics, 100 (2006), 043521. doi: 10.1063/1.2234824. Google Scholar [28] M. Gardel, J. H. Shin, L. Mahadevan, F. C. MacKintosh, P. Matsudaira and D. A. Weitz, Elastic behavior of cross-linked and bundled actin networks,, Science, 304 (2004), 1301. doi: 10.1126/science.1095087. Google Scholar [29] E. F Gramsbergen, L. Longa and W. H. de Jeu, Landau theory of nematic isotropic phase transitions,, Phys. Rep., 135 (1986), 195. doi: 10.1016/0370-1573(86)90007-4. Google Scholar [30] P. Hajlasz, Sobolev mappings, co-area formula, and related topics,, Proceedings on Analysis and Geometry, (2000), 227. Google Scholar [31] I. Kundler and H. Finkelmann, Strain-induced director reorientation in nematic liquid single crystal elastomers,, Macromolecular Rapid Communications, 16 (1995), 679. doi: 10.1002/marc.1995.030160908. Google Scholar [32] C Luo, Modeling, Analysis and Numerical Simulations of Liquid Crystal Elastomers,, Ph.D dissertation, (2010). Google Scholar [33] W. Maier and A. Saupe, A simple molecular statistical theory of the nematic liquid crystalline-liquid phase,, I Z Naturf. a, 14 (1959), 882. Google Scholar [34] A. Majumdar, Equilibrium order parameters of nematic liquid crystals in the landau-de gennes theory,, European J. Appl. Math., 21 (2010), 181. doi: 10.1017/S0956792509990210. Google Scholar [35] A. Majumdar, The landau-de gennes theory of nematic liquid crystals: Uniaxiality versus biaxiality,, Communications on Pure and Applied Analysis, 11 (2012), 1303. doi: 10.3934/cpaa.2012.11.1303. Google Scholar [36] A. Majumdar and A. Zarnescu, Landau-de gennes theory of nematic liquid crystals: The oseen-frank limit and beyond,, Archive for rational mechanics and analysis, 196 (2010), 227. doi: 10.1007/s00205-009-0249-2. Google Scholar [37] M. Marcus and V. J. Mizel, Transformations by functions in sobolev spaces and lower semicontinuity for parametric variational problems,, Bulletin of the American Mathematical Society, 79 (1973), 790. doi: 10.1090/S0002-9904-1973-13319-1. Google Scholar [38] N. J. Mottram and C. Newton, Introduction to Q-Tensor Theory,, in Research Report no. 10, (2004). Google Scholar [39] A. Rapini and M. Papoular, Surface anchoring of nematic liqud crystals,, J. Physique Coll, 30 (1969), 54. Google Scholar [40] A. Sanchez-Ferrer and H. Finkelmann, Uniaxial and shear deformation in smectic-c main-chain liquid-crystalline elastomers,, Macromolecules, 41 (2008), 970. doi: 10.1021/ma7025644. Google Scholar [41] A. Sanchez-Ferrer and H. Finkelmann, Mechanical properties of new main-chain liquid-crystalline elastomers,, Mol. Cryst. Liq. Cryst., 508 (2009), 348. doi: 10.1080/15421400903065861. Google Scholar [42] B. Wagner, R. Tharmann, I. Haase, M. Fischer and A. R. Bausch, Cytoskeletal polymer networks: The molecule structure of cross-linkers determines macroscopic properties of cochlear outer hair-cells,, Proc.Natl. Acad. Sci USA, 103 (2006), 13974. Google Scholar [43] M. Warner and E. M. Terentjev, Liquid Crystal Elastomers,, Oxford University Press, (2007). Google Scholar [44] E. R. Zubarev, S. A. Kuptsov, T. I. Yuranova, R. V. Talroze and H. Finkelmann, Monodomain liquid crystalline networks: Reorientation mechanism from uniform to stripe domains,, Liquid crystals, 26 (1999), 1531. doi: 10.1080/026782999203869. Google Scholar

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##### References:
 [1] V. Agostiniani and A. DeSimone, Ogden-type energies for nematic elastomers,, Int J. Nonlinear Mechanics, 47 (2012), 402. Google Scholar [2] D. Anderson, D. Carlson and E. Fried, A continuum-mechanical theory for nematic elastomers,, J. Elasticity, 56 (1999), 33. doi: 10.1023/A:1007647913363. Google Scholar [3] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity,, Archive for Rational Mechanics and Analysis, 63 (1977), 337. doi: 10.1007/BF00279992. Google Scholar [4] J. M. Ball, Global invertibility of sobolev functions and the interpenetration of matter,, Proceedings Royal Soc. Edinburgh A, 88 (1981), 315. doi: 10.1017/S030821050002014X. Google Scholar [5] J. M. Ball and A. Majumdar, Nematic liquid crystals: From maier-saupe to a continuum theory,, Molecular Crystals and Liquid Crystals, 525 (2010), 1. doi: 10.1080/15421401003795555. Google Scholar [6] M. Barchiesi and A. DeSimone, Frank energy for nematic elastomers. A nonlinear model,, Preprint, (2013). Google Scholar [7] M. C. Calderer and B. Mukherjee, Chevron patterns in liquid crystal flows,, Physica D, 98 (1996), 201. doi: 10.1016/0167-2789(96)00051-6. Google Scholar [8] M. C. Calderer and B. Mukherjee, On poiseuille flow of liquid crystals,, Liquid Crystals, 22 (1997), 121. doi: 10.1080/026782997209487. Google Scholar [9] M. C. Calderer and B. Mukherjee, Mathematical issues in the modeling of flow behavior of polymeric liquid crystals,, J. Rheol., 42 (1998), 1519. Google Scholar [10] M. C. Calderer, Critical size of stripe patterns in liquid crystal elastomers under extension,, Preprint, (2013). Google Scholar [11] M. C. Calderer, C. A. Garavito and C. Luo, Liquid Crystal Elastomers and Phase Transitions in Rod Networks,, Technical Report, (2013). Google Scholar [12] M. C. Calderer and C. Liu, Liquid crystal flow: dynamic and static configurations,, SIAM Journal on Applied Mathematics, 20 (2000), 1225. Google Scholar [13] M. C. Calderer, C. Liu and B. Yan, A mathematical theory for nematic elastomers with non-uniform prolate spheroids,, in Advances in Applied and Computational Mathematics (eds. F. Liu, (2006), 245. Google Scholar [14] M. C. Calderer and C. Luo, Numerical study of liquid crystal elastomers by a mixed finite element method,, European J. Appl. Math., 23 (2012), 121. doi: 10.1017/S0956792511000313. Google Scholar [15] P. Cesana and A. DeSimone, Strain-order coupling in nematic elastomers: Equilibrium configurations,, Math. Models Methods Appl. Sci, 19 (2009), 601. doi: 10.1142/S0218202509003541. Google Scholar [16] P. Cesana and A. DeSimone, Quasiconvex envelopes of energies for nematic elastomers in the small strain regime and applications,, J. Mech. Phys. Solids, 59 (2011), 787. doi: 10.1016/j.jmps.2011.01.007. Google Scholar [17] P-G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity,, Arch. Rational Mech. Anal., 97 (1987), 171. doi: 10.1007/BF00250807. Google Scholar [18] S. Conti, A. DeSimone and G. Dolzmann, Soft elastic response of stretched sheets of nematic elastomers: A numerical study,, Journal of the Mechanics and Physics of Solids, 50 (2002), 1431. doi: 10.1016/S0022-5096(01)00120-X. Google Scholar [19] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals,, Oxford University Press, (1995). Google Scholar [20] A. DeSimone and G. Dolzmann, Material instabilities in nematic elastomers,, Physica D, 136 (2000), 175. doi: 10.1016/S0167-2789(99)00153-0. Google Scholar [21] A. DeSimone and G. Dolzmann, Macroscopic response of nematic elastomers via relaxation of a class of $SO(3)$-invariant energies,, Arch. Rat. Mech. Anal., 161 (2002), 181. doi: 10.1007/s002050100174. Google Scholar [22] A. DeSimone and L. Teresi, Elastic energies for nematic elastomers,, The European Physical Journal E: Soft Matter and Biological Physics, 29 (2009), 191. doi: 10.1140/epje/i2009-10467-9. Google Scholar [23] J. L. Ericksen, Liquid crystals with variable degree of orientation,, Archive for Rational Mechanics and Analysis, 113 (1991), 97. doi: 10.1007/BF00380413. Google Scholar [24] I. Fonseca and W. Gangbo, Local invertibility of sobolev functions,, SIAM J. Math. Anal., 26 (1995), 280. doi: 10.1137/S0036141093257416. Google Scholar [25] G. Forest, Q. Wang and R. Zhou, A kinetic theory for solutions of nonhomogeneous nematic liquid crystalline polymers with density variations,, J.Fluid Engineering, 126 (2004), 180. Google Scholar [26] E. Fried and S. Sellers, Free-energy density functions for nematic elastomers,, J. Mech. Phys. Solids, 52 (1999), 1671. doi: 10.1016/j.jmps.2003.12.005. Google Scholar [27] E. Fried and S. Sellers, Soft elasticity is not necessary for striping in nematic elastomers,, Journal of Applied Physics, 100 (2006), 043521. doi: 10.1063/1.2234824. Google Scholar [28] M. Gardel, J. H. Shin, L. Mahadevan, F. C. MacKintosh, P. Matsudaira and D. A. Weitz, Elastic behavior of cross-linked and bundled actin networks,, Science, 304 (2004), 1301. doi: 10.1126/science.1095087. Google Scholar [29] E. F Gramsbergen, L. Longa and W. H. de Jeu, Landau theory of nematic isotropic phase transitions,, Phys. Rep., 135 (1986), 195. doi: 10.1016/0370-1573(86)90007-4. Google Scholar [30] P. Hajlasz, Sobolev mappings, co-area formula, and related topics,, Proceedings on Analysis and Geometry, (2000), 227. Google Scholar [31] I. Kundler and H. Finkelmann, Strain-induced director reorientation in nematic liquid single crystal elastomers,, Macromolecular Rapid Communications, 16 (1995), 679. doi: 10.1002/marc.1995.030160908. Google Scholar [32] C Luo, Modeling, Analysis and Numerical Simulations of Liquid Crystal Elastomers,, Ph.D dissertation, (2010). Google Scholar [33] W. Maier and A. Saupe, A simple molecular statistical theory of the nematic liquid crystalline-liquid phase,, I Z Naturf. a, 14 (1959), 882. Google Scholar [34] A. Majumdar, Equilibrium order parameters of nematic liquid crystals in the landau-de gennes theory,, European J. Appl. Math., 21 (2010), 181. doi: 10.1017/S0956792509990210. Google Scholar [35] A. Majumdar, The landau-de gennes theory of nematic liquid crystals: Uniaxiality versus biaxiality,, Communications on Pure and Applied Analysis, 11 (2012), 1303. doi: 10.3934/cpaa.2012.11.1303. Google Scholar [36] A. Majumdar and A. Zarnescu, Landau-de gennes theory of nematic liquid crystals: The oseen-frank limit and beyond,, Archive for rational mechanics and analysis, 196 (2010), 227. doi: 10.1007/s00205-009-0249-2. Google Scholar [37] M. Marcus and V. J. Mizel, Transformations by functions in sobolev spaces and lower semicontinuity for parametric variational problems,, Bulletin of the American Mathematical Society, 79 (1973), 790. doi: 10.1090/S0002-9904-1973-13319-1. Google Scholar [38] N. J. Mottram and C. Newton, Introduction to Q-Tensor Theory,, in Research Report no. 10, (2004). Google Scholar [39] A. Rapini and M. Papoular, Surface anchoring of nematic liqud crystals,, J. Physique Coll, 30 (1969), 54. Google Scholar [40] A. Sanchez-Ferrer and H. Finkelmann, Uniaxial and shear deformation in smectic-c main-chain liquid-crystalline elastomers,, Macromolecules, 41 (2008), 970. doi: 10.1021/ma7025644. Google Scholar [41] A. Sanchez-Ferrer and H. Finkelmann, Mechanical properties of new main-chain liquid-crystalline elastomers,, Mol. Cryst. Liq. Cryst., 508 (2009), 348. doi: 10.1080/15421400903065861. Google Scholar [42] B. Wagner, R. Tharmann, I. Haase, M. Fischer and A. R. Bausch, Cytoskeletal polymer networks: The molecule structure of cross-linkers determines macroscopic properties of cochlear outer hair-cells,, Proc.Natl. Acad. Sci USA, 103 (2006), 13974. Google Scholar [43] M. Warner and E. M. Terentjev, Liquid Crystal Elastomers,, Oxford University Press, (2007). Google Scholar [44] E. R. Zubarev, S. A. Kuptsov, T. I. Yuranova, R. V. Talroze and H. Finkelmann, Monodomain liquid crystalline networks: Reorientation mechanism from uniform to stripe domains,, Liquid crystals, 26 (1999), 1531. doi: 10.1080/026782999203869. Google Scholar
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