Landau–de Gennes theory
In physics, Landau–de Gennes theory describes the NI transition, i.e., phase transition between nematic liquid crystals and isotropic liquids, which is based on the classical Landau's theory and was developed by Pierre-Gilles de Gennes in 1969.[1][2] The phenomonological theory uses the tensor as an order parameter in expandiing the free energy density.[3][4]
Mathematical description
The NI transition is a first-order phase transition, albeit it is very weak. The order parameter is the tensor, which is symmetric, traceless, second-order tensor and vanishes in the isotripic liquid phase. We shall consider a uniaxial tensor, which is defined by
where is the scalar order parameter and is the director. The tensor is zero in the isotropic liquid phase since the scalar order parameter is zero, but becomes non-zero in the nematic phase.
Near the NI transition, the (Helmholtz or Gibbs) free energy density is expanded about as
or more compactly
Further, we can expand , and with being three positive constants. Now substituting the tensor results in[5]
This is minimized when
The two required solutions of this equation are
The NI transition temperature is not simply equal to (which would be the case in second-order phase transition), but is given by
is the scalar order parameter at the transition.
References
- ^ De Gennes, P. G. (1969). Phenomenology of short-range-order effects in the isotropic phase of nematic materials. Physics Letters A , 30 (8), 454-455.
- ^ De Gennes, P. (1971). Short range order effects in the isotropic phase of nematics and cholesterics. Molecular Crystals and Liquid Crystals, 12(3), 193-214.
- ^ De Gennes, P. G., & Prost, J. (1993). The physics of liquid crystals (No. 83). Oxford university press.
- ^ Mottram, N. J., & Newton, C. J. (2014). Introduction to Q-tensor theory. arXiv preprint arXiv:1409.3542.
- ^ Kleman, M., & Lavrentovich, O. D. (Eds.). (2003). Soft matter physics: an introduction. New York, NY: Springer New York.