June  2011, 4(3): 565-579. doi: 10.3934/dcdss.2011.4.565

Isotropic-nematic phase transitions in liquid crystals

1. 

Department of Mathematics, Piazza di Porta S. Donato 5, 40127-Bologna

2. 

Dipartimento di Matematica, Via Valotti 9, 25133 Brescia, Italy

3. 

DIBE, Via Opera Pia 11a, 16145 Genova, Italy

Received  April 2009 Revised  August 2009 Published  November 2010

The paper derives the evolution equations for a nematic liquid crystal, under the action of an electromagnetic field, and characterizes the transition between the isotropic and the nematic state. The non-simple character of the continuum is described by means of the director, of the degree of orientation and their space and time derivatives. Both the degree of orientation and the director are regarded as internal variables and their evolution is established by requiring compatibility with the second law of thermodynamics. As a result, admissible forms of the evolution equations are found in terms of appropriate terms arising from a free-enthalpy potential. For definiteness a free-enthalpy is then considered which provides directly the dielectric and magnetic anisotropies. A characterization is given of thermally-induced transitions with the degree of orientation as a phase parameter.
Citation: Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Isotropic-nematic phase transitions in liquid crystals. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 565-579. doi: 10.3934/dcdss.2011.4.565
References:
[1]

P. G. de Gennes and J. Prost, "The Physics of Liquid Crystals," Clarendon, Oxford, 1998.

[2]

S. Singh, Phase transitions in liquid crystals, Phys. Rep., 324 (2000), 107-269.

[3]

J. L. Ericksen, Liquid crystals with variable degree of orientation, Arch. Rational Mech. Anal., 113 (1991), 97-120.

[4]

C.-P. Fan, Second order phase transitions in liquid crystals, Chin. J. Phys., 12 (1974), 24-31.

[5]

C.-P. Fan and M. J. Stephen, Isotropic-nematic phase transition in liquid crystal, Phys. Rev. Letters, 25 (1970), 500-503.

[6]

M. C. Calderer, Stability of shear flows of polimeric liquid crystals, J. Non-Newtonian Fluid Mech., 43 (1992), 351-368.

[7]

F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), 265-283.

[8]

P. Cermelli, E. Fried and M. E. Gurtin, Sharp-interface nematic-isotropic phase transitions without flow, Arch. Rational Mech. Anal., 174 (2004), 151-178.

[9]

D. R. Anderson, D. E. Carlson and E. Fried, A continuum-mechanical theory for nematic elastomers, J. Elasticity, 56 (1999), 33-58.

[10]

P. Cermelli and E. Fried, The evolution equation for a disclination in a nematic liquid crystal, Proc. R. Soc. Lond. - A, 458 (2002), 1-20,

[11]

P. Biscari, G. Napoli and S. Turzi, Bulk and surface biaxiality in nematic liquid crystals, Phys. Rev. - E, 74 (2006), 031708-7. doi: doi:10.1103/PhysRevE.74.031708.

[12]

P. Biscari, M. C. Calderer and E. M. Terentjev, Landau-de Gennes theory of isotropic-nematic-smectic liquid crystal transitions, Phys. Rev. - E, 75 (2007), 051707-11. doi: doi:10.1103/PhysRevE.75.051707.

[13]

M. Fabrizio, C. Giorgi and A. Morro, A thermodynamic approach to non-isothermal phase-field evolution in continuum physics, Physica - D, 214 (2006), 144-156.

show all references

References:
[1]

P. G. de Gennes and J. Prost, "The Physics of Liquid Crystals," Clarendon, Oxford, 1998.

[2]

S. Singh, Phase transitions in liquid crystals, Phys. Rep., 324 (2000), 107-269.

[3]

J. L. Ericksen, Liquid crystals with variable degree of orientation, Arch. Rational Mech. Anal., 113 (1991), 97-120.

[4]

C.-P. Fan, Second order phase transitions in liquid crystals, Chin. J. Phys., 12 (1974), 24-31.

[5]

C.-P. Fan and M. J. Stephen, Isotropic-nematic phase transition in liquid crystal, Phys. Rev. Letters, 25 (1970), 500-503.

[6]

M. C. Calderer, Stability of shear flows of polimeric liquid crystals, J. Non-Newtonian Fluid Mech., 43 (1992), 351-368.

[7]

F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), 265-283.

[8]

P. Cermelli, E. Fried and M. E. Gurtin, Sharp-interface nematic-isotropic phase transitions without flow, Arch. Rational Mech. Anal., 174 (2004), 151-178.

[9]

D. R. Anderson, D. E. Carlson and E. Fried, A continuum-mechanical theory for nematic elastomers, J. Elasticity, 56 (1999), 33-58.

[10]

P. Cermelli and E. Fried, The evolution equation for a disclination in a nematic liquid crystal, Proc. R. Soc. Lond. - A, 458 (2002), 1-20,

[11]

P. Biscari, G. Napoli and S. Turzi, Bulk and surface biaxiality in nematic liquid crystals, Phys. Rev. - E, 74 (2006), 031708-7. doi: doi:10.1103/PhysRevE.74.031708.

[12]

P. Biscari, M. C. Calderer and E. M. Terentjev, Landau-de Gennes theory of isotropic-nematic-smectic liquid crystal transitions, Phys. Rev. - E, 75 (2007), 051707-11. doi: doi:10.1103/PhysRevE.75.051707.

[13]

M. Fabrizio, C. Giorgi and A. Morro, A thermodynamic approach to non-isothermal phase-field evolution in continuum physics, Physica - D, 214 (2006), 144-156.

[1]

Pavel Drábek, Stephen Robinson. Continua of local minimizers in a quasilinear model of phase transitions. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 163-172. doi: 10.3934/dcds.2013.33.163

[2]

Sylvie Benzoni-Gavage, Laurent Chupin, Didier Jamet, Julien Vovelle. On a phase field model for solid-liquid phase transitions. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 1997-2025. doi: 10.3934/dcds.2012.32.1997

[3]

Valeria Berti, Mauro Fabrizio, Diego Grandi. A phase field model for liquid-vapour phase transitions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 317-330. doi: 10.3934/dcdss.2013.6.317

[4]

Chun Liu. Dynamic theory for incompressible Smectic-A liquid crystals: Existence and regularity. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 591-608. doi: 10.3934/dcds.2000.6.591

[5]

Fanghua Lin, Chun Liu. Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete and Continuous Dynamical Systems, 1996, 2 (1) : 1-22. doi: 10.3934/dcds.1996.2.1

[6]

Alice Fiaschi. Rate-independent phase transitions in elastic materials: A Young-measure approach. Networks and Heterogeneous Media, 2010, 5 (2) : 257-298. doi: 10.3934/nhm.2010.5.257

[7]

Jiayan Yang, Dongpei Zhang. Superfluidity phase transitions for liquid $ ^{4} $He system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 5107-5120. doi: 10.3934/dcdsb.2019045

[8]

Wenya Ma, Yihang Hao, Xiangao Liu. Shape optimization in compressible liquid crystals. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1623-1639. doi: 10.3934/cpaa.2015.14.1623

[9]

Marita Thomas, Sven Tornquist. Discrete approximation of dynamic phase-field fracture in visco-elastic materials. Discrete and Continuous Dynamical Systems - S, 2021, 14 (11) : 3865-3924. doi: 10.3934/dcdss.2021067

[10]

Claude Vallée, Camelia Lerintiu, Danielle Fortuné, Kossi Atchonouglo, Jamal Chaoufi. Modelling of implicit standard materials. Application to linear coaxial non-associated constitutive laws. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1641-1649. doi: 10.3934/dcdss.2013.6.1641

[11]

Honghu Liu. Phase transitions of a phase field model. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 883-894. doi: 10.3934/dcdsb.2011.16.883

[12]

Yuming Chu, Yihang Hao, Xiangao Liu. Global weak solutions to a general liquid crystals system. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2681-2710. doi: 10.3934/dcds.2013.33.2681

[13]

Carlos J. García-Cervera, Sookyung Joo. Reorientation of smectic a liquid crystals by magnetic fields. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 1983-2000. doi: 10.3934/dcdsb.2015.20.1983

[14]

Jinhae Park, Feng Chen, Jie Shen. Modeling and simulation of switchings in ferroelectric liquid crystals. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1419-1440. doi: 10.3934/dcds.2010.26.1419

[15]

Shaoqiang Tang, Huijiang Zhao. Stability of Suliciu model for phase transitions. Communications on Pure and Applied Analysis, 2004, 3 (4) : 545-556. doi: 10.3934/cpaa.2004.3.545

[16]

Tatyana S. Turova. Structural phase transitions in neural networks. Mathematical Biosciences & Engineering, 2014, 11 (1) : 139-148. doi: 10.3934/mbe.2014.11.139

[17]

Boling Guo, Yongqian Han, Guoli Zhou. Random attractor for the 2D stochastic nematic liquid crystals flows. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2349-2376. doi: 10.3934/cpaa.2019106

[18]

Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211

[19]

Geng Chen, Ping Zhang, Yuxi Zheng. Energy conservative solutions to a nonlinear wave system of nematic liquid crystals. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1445-1468. doi: 10.3934/cpaa.2013.12.1445

[20]

Xian-Gao Liu, Jie Qing. Globally weak solutions to the flow of compressible liquid crystals system. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 757-788. doi: 10.3934/dcds.2013.33.757

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (83)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]