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
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
tensor, which is symmetric, traceless, second-order tensor and vanishes in the isotropic 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
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
is expanded about as
:
or more compactly
:
where
are functions of temperature. Near the phase transition, we can expand
,
and
with
being three positive constants. Now substituting the
tensor results in
[Kleman, M., & Lavrentovich, O. D. (Eds.). (2003). Soft matter physics: an introduction. New York, NY: Springer New York.]
:
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
{{reflist, 30em
Soft matter
Phase transitions
Liquid crystals