Landau–de Gennes Theory
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In physics, Landau–de Gennes theory describes the NI transition, i.e., phase transition between nematic
liquid crystals Liquid crystal (LC) is a state of matter whose properties are between those of conventional liquids and those of solid crystals. For example, a liquid crystal can flow like a liquid, but its molecules may be oriented in a common direction as i ...
and isotropic liquids, which is based on the classical Landau's theory and was developed by
Pierre-Gilles de Gennes Pierre-Gilles de Gennes (; 24 October 1932 – 18 May 2007) was a French physicist and the Nobel Prize laureate in physics in 1991. Education and early life He was born in Paris, France, and was home-schooled to the age of 12. By the age of ...
in 1969. The phenomonological theory uses the \mathbf tensor as an
order parameter In physics, chemistry, and other related fields like biology, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic s ...
in expanding the free energy density.


Mathematical description

The NI transition is a first-order phase transition, albeit it is very weak. The order parameter is the \mathbf tensor, which is symmetric, traceless, second-order tensor and vanishes in the isotropic liquid phase. We shall consider a uniaxial \mathbf Q tensor, which is defined by :\mathbf Q = S(\mathbf n\otimes\mathbf n - \tfrac\mathbf I) where S=S(T) is the scalar order parameter and \mathbf n is the director. The \mathbf Q tensor is zero in the isotropic liquid phase since the scalar order parameter S is zero, but becomes non-zero in the nematic phase. Near the NI transition, the (
Helmholtz Hermann Ludwig Ferdinand von Helmholtz (; ; 31 August 1821 – 8 September 1894; "von" since 1883) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The ...
or Gibbs) free energy density \mathcal is expanded about as :\mathcal = \mathcal_0 + \frac Q_Q_ - \frac Q_Q_Q_ + \frac (Q_Q_)^2 or more compactly :\mathcal = \mathcal_0 + \frac\mathrm \,\mathbf^2 - \frac\mathrm \,\mathbf^3 + \frac(\mathrm \,\mathbf^2)^2 where (A,B,C) are functions of temperature. Near the phase transition, we can expand A(T)=a (T-T_*)+\cdots, B(T) = b + \cdots and C(T)=c + \cdots with (a,b,c) being three positive constants. Now substituting the \mathbf Q tensor results inKleman, M., & Lavrentovich, O. D. (Eds.). (2003). Soft matter physics: an introduction. New York, NY: Springer New York. :\mathcal - \mathcal_0 = \frac(T-T_*)S^2 - \frac S^3 + \fracS^4. This is minimized when :3a(T-T_*) S - b S^2 + 2c S^3=0. The two required solutions of this equation are :\begin\text & \,\,S_I = 0,\\ \text & \,\,S_N = \frac \left +\sqrt\,\right0. \end The NI transition temperature T_ is not simply equal to T_* (which would be the case in second-order phase transition), but is given by :T_ = T_* + \frac, \quad S_ = \frac S_ is the scalar order parameter at the transition.


References

{{reflist, 30em Soft matter Phase transitions Liquid crystals