Therefore for the case of a chiral liquid crystal, the total free energy density is given by:
2.
A fourth term is also commonly added to the Frank free energy density called the saddle-splay energy that describes the surface interaction.
3.
Where M _ { \ eta } is the mobility, f is the free energy density, and \ eta is the nonconserved order parameter.
4.
If inclusions are added to a liquid crystal, an additional term contributes to the free energy density due to their presence, often characterized by a term known as the Rapini approximation:
5.
Since the-\ frac { H ^ 2 \ chi _ \ perp } { 2 } term is spatially invariant, it can be ignored and so the magnetic contribution to the distortion free energy density becomes:
6.
To understand the effect a magnetic field produces on the distortion free energy density, a small region of local nematic order \ mathbf { \ hat { n } } is often considered in which \ chi _ \ perp and \ chi _ \ parallel is the magnetic susceptibility perpendicular and parallel to \ mathbf { \ hat { n } }.
7.
Where \ mathcal { F } _ { T } is the total free energy density of a liquid crystal, \ mathcal { F } _ { 0 } is the free energy density associated with a uniformly aligned nematic, and \ mathcal { F } _ { d } is the contribution to the free energy density due to distortions in this order.
8.
Where \ mathcal { F } _ { T } is the total free energy density of a liquid crystal, \ mathcal { F } _ { 0 } is the free energy density associated with a uniformly aligned nematic, and \ mathcal { F } _ { d } is the contribution to the free energy density due to distortions in this order.
9.
Where \ mathcal { F } _ { T } is the total free energy density of a liquid crystal, \ mathcal { F } _ { 0 } is the free energy density associated with a uniformly aligned nematic, and \ mathcal { F } _ { d } is the contribution to the free energy density due to distortions in this order.
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