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] The phenomonologivcal theory uses the tensor as an order parameter in expandiing the free energy density.[2][3]

Mathematical description

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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 vasbishes in the isotripic liquid phase. We shall consider a uniaxial   tensor, which is defined by

 

where   is the scalar order parameter. 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[4]

 

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

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  1. ^ 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.
  2. ^ De Gennes, P. G., & Prost, J. (1993). The physics of liquid crystals (No. 83). Oxford university press.
  3. ^ Mottram, N. J., & Newton, C. J. (2014). Introduction to Q-tensor theory. arXiv preprint arXiv:1409.3542.
  4. ^ Kleman, M., & Lavrentovich, O. D. (Eds.). (2003). Soft matter physics: an introduction. New York, NY: Springer New York.