15.2 Isotropic↔Nematic Phase Transition 277
15.2 Isotropic$Nematic Phase Transition.
15.2.1 Order Parameter Tensor.
The existence of a non-zero second rank alignment tensor in thermal equilibrium,
distinguishes the nematic phase of a liquid crystal from its isotropic liquid state. The
order parameter tensor can be introduced phenomenologically via the anisotropic, i.e.
symmetric traceless part of the electric or magnetic susceptibility tensor or, and this
is preferred here, microscopically as an average over the orientational distribution
function. As in Sect.12.2.2, particles with a symmetry axis parallel to the unit vector
uare considered. In most nematics, the orientational distributionf = f(u)does
not depend on the sign ofu, even when the particles do have a polar character. The
propertyf(u)= f(−u)is referred to ashead-tail symmetry. The lowest moment
which distinguishes an anisotropic distribution from an isotropic one is thesecond
rank alignment tensor,cf.(12.14) and (12.17),
aμν=〈φμν〉,φμν=ζ 2 uμuν,ζ 2 =√
15
2
. (15.1)
The bracket indicates the average over the orientational distribution, viz.
〈...〉=
∫
...f(u)d^2 u.The quadrupole moment tensor, cf. (10.29) has the same symmetry as the second
rank alignment tensoraμν∼〈uμuν〉. Therefore, the tensor〈uμuν〉is also denoted
byQμνand calledQ-tensor, [67].
The general properties of symmetric second rank tensors discussed in Chap. 5
apply to the traceless tensoraμνdefined in (15.1). In particular, in a principal axes
frame with the principal axes parallel to the mutually perpendicular unit vectorse(i),
i= 1 , 2 ,3, the tensor is expressed as,
aμν=√
3
2
a 0 e(μ^3 )e(ν^3 )+√
2
2
a 1(
e(μ^1 )eν(^1 )−e(μ^2 )e(ν^2 ))
. (15.2)
The factors in front of the quantitiesa 0 anda 1 have been chosen such that the
magnitude of the alignment tensor, viz. the second scalar invariantI 2 =aμνaμνis
equal toa^20 +a^21. For a comparison with the relations given in Sect.5.2.4, notice
that the quantities corresponding toS ̄,sandqof (5.11)are0,
√
3
2 a^0 anda^1 /√
- By
analogy to (5.13), the principal valuesa(i),i= 1 , 2 ,3 of the alignment tensor are
a(^1 )=−1
√
6
a 0 +1
√
2
a 1 , a(^2 )=−1
√
6
a 0 −1
√
2
a 1 , a(^3 )=√
2
3
a 0. (15.3)