Tensors for Physics

294 15 Liquid Crystals and Other Anisotropic Fluids

Fig. 15.7Tetrahedron
embedded within a cube.
Thelinesconnecting the
center with the four corners
of the tetrahedron show the
directions of the vectorsui

be four unit vectors pointing from the center to the corners of the tetrahedron. In this
case, the orientational order is specified by a third rank tensorTμνλdefined by

Tμνλ=ζ 3

〈 4

∑

i= 1

uiμuiνuiλ

〉

∼

〈

eμxeyνezλ

〉

, (15.50)

whereζ 3 is a numerical factor which can be chosen conveniently. The corners of the
tetrahedron can be placed on the corners of a cube, cf. Fig.15.7. The second relation
in (15.50) involves the body fixed unit vectorsex,eyandezwhich are parallel to the
axes of this cube.
The tetradic third rank order parameter tensorT has negative parity, just like a
first rank dipolar order. This is in contradistinction to the second and fourth rank
order parameter tensors, which have positive parity. As first pointed out in [94], the
third rank order parameter is needed for the description of ‘banana phases’ of liquid
crystals which are composed of particles with a bent core.

15.5 Energetic Coupling of Order Parameter Tensors

Sometimes, more than one order parameter tensor is needed to describe the properties
of a substance and the relevant phenomena. In general, the pertaining free energy
containscouplingterms,whosestructuredependsontheranksofthetensorsinvolved.
The case of second and fourth rank tensors was already discussed in Sect.15.3.3.
Three other examples, viz. the coupling of two second rank tensors, of a second rank
tensor with a vector and with a third rank tensor are presented here.

15.5.1 Two Second Rank Tensors.

Letaandbbe two symmetric traceless second rank tensors which describe the
orientational properties of a substance. Examples are the alignment tensor associated
withsidegroupsandwiththebackboneofaside-chainpolymer,asstudiedin[95, 96],