# American Institute of Mathematical Sciences

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.
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