15.2 Isotropic↔Nematic Phase Transition 281
has to be obeyed at equilibrium, these relations imply
a=ani≡2 B
3 C
, Ani=A 0 ( 1 −T∗/Tni)=1
3
Bani=2 B^2
9 C
. (15.17)
Typical thermotropic nematic liquid crystals have an order parameterS 2 of about
0 .4, at the transition temperature. This corresponds toani= 0. 9 ≈1. The relative
difference between the transition temperatureTniandT∗,viz.
δni=(Tni−T∗)/Tni= 2 B^2 /( 9 A 0 C)=ani^2 C/( 2 B 0 ), (15.18)is of the order 10−^2. The Exercise15.1provides a derivation of the potential function
(15.12) and it yields specific values for the coefficientsA 0 ,B,C. In the literature,
variables referring to the nematic-isotropic phase transition are also labelled with the
letter “K”, rather than “ni”, likeTKoraKinstead ofTniandani. The letter “K” stems
from “Klärpunkt”, meaning “clearing point”. The reason is: polycrystalline liquid
crystals are turbid and they become clear in the isotropic phase.
It is convenient to introduce the scaled alignment tensora∗μν, a reduced potential
Φ∗and a reduced relative temperatureθvia
aμν=aniaμν∗,Φ=ani^2 AniΦ∗, (15.19)θ=A(T)/Ani=( 1 −T∗/T)/( 1 −T∗/Tni)=(Tni/T)(T−T∗)/(Tni−T∗).
(15.20)Then the resulting expressions for the scaled Landau de Gennes potential
(ΦLdG)∗=1
2
θa∗μνa∗νμ−√
6 aμν∗aνκ∗a∗κμ+1
2
(a∗μνa∗νμ)^2 , (15.21)is universal in the sense that the original coefficientsA 0 ,B,Cno longer show up
explicitly. Witha=ania∗, the scaled expression corresponding to (15.14)is
(ΦLdG)∗=1
2
θ(a∗)^2 −(a∗)^3 +1
2
(a∗)^4. (15.22)The transition temperatureTniand the temperatureT∗correspond toθ=1 and
θ=0, respectively. The relation corresponding to (15.15)is
a∗(θ− 3 a∗+ 2 (a∗)^2 )= 0.The resulting equilibrium value of the order parameter in the nematic phase is
aeq∗ =3
4
+
1
4
√
9 − 8 θ, θ≤