296 15 Liquid Crystals and Other Anisotropic Fluids
15.5.2 Second-Rank Tensor and Vector.
Letd∼〈e〉anda∼〈uu〉>be a polar vector and a symmetric traceless second
rank alignment tensor which describe the orientational properties of a substance.
Hereeis a unit vector parallel to a molecular electric dipole moment which need not
be parallel tou. The electric polarizationPis proportional tod. The dimensionless
free energyΦ=Φ(d,a)is written as
Φ=Φd+Φa+Φda,Φda=−c 1 dμ∇νaνμ+1
2
c 2 dμdνaμν, (15.54)whereΦa =Φa(a)is a Landau-de Gennes potential function,Φd=Φd(d)is
a similar expression for the vectord, andΦda=Φda(d,a), with the coefficients
c 1 ,c 2 characterizes the coupling between the vector and the tensor. Terms of higher
order are possible, but not included here, for simplicity. A scalar linear in bothd
andamust involve an additional vector, here it is the nabla-vector. The coupling
coefficientc 1 is a true scalar whendis a polar vector. The corresponding expression
for an axial vector must contain a coefficientc 1 which is a pseudo-scalar in order to
conserve parity. The coefficientc 2 is a true scalar, in any case.
The derivatives of the potential with respect to the vector and to the tensor are
∂Φ
∂dμ=Φμd−c 1 ∇νaνμ+c 2 dνaνμ,∂Φ
∂aμν=Φμνa +c 1 ∇μdν+1
2
c 2 dμdν.
(15.55)In thermal equilibrium, these derivatives are equal to zero. For the special case where
Φd=^12 dμdμapplies, one obtains
dμ+c 2 dνaνμ=c 1 ∇νaνμ. (15.56)This relation underlies theflexo-electric effect, viz. an electric polarizationPcaused
by spatial derivatives of the director fieldn, in nematic liquid crystals. The phenom-
enological description of this effect is [67]
P=e 1 n∇·n+e 3 (∇×n)×n,which, due tonν∇μnν=0, is equivalent to
Pμ=e 1 nμ∇νnν+e 3 nν∇νnμ. (15.57)The phenomenological coefficientse 1 ande 3 characterize the electric polarization
caused by splay and by bend deformations, cf. Sect.15.3.1. For the uniaxial alignment
aμν=
√
3
2 aeqnμnνwith the equilibrium order parameteraeq, an expression of the
form (15.57) is obtained from (15.56) with the help of the relationPμ=Prefdμ.
HerePrefis a reference value for the electric polarization which is proportional to