292 15 Liquid Crystals and Other Anisotropic Fluids
15.4.2 Landau Theory for the Isotropic-Cubatic
Phase Transition
A phenomenological theory for the phase transition of an isotropic state to one
with cubic symmetry can be made in analogy to the isotropic-nematic transition, cf.
Sect.15.2.2. Such an approach was first proposed independently, in [90] and [91]
for cubic crystals. Notice, however, that the long range positional order typical for
crystalline solids is not treated explicitly in this theory which focuses on the bond
orientational order. To stress this point, the ordered state as treated here, is referred to
as “cubatic”, be it an ordered fluid like a smectic D liquid crystal, a fluid containing
oriented cubic particles, or a true cubic crystal.
By analogy with the Landau-de Gennes theory for nematics, cf. Sect.15.2.2,
a dimensionless free energy potential is formulated:
Φ=ΦL≡
1
2
Aaμνλκaμνλκ−1
3
√
30 Baμνλκaλκσ τaστμν+1
4
C(aμνλκaμνλκ)^2 ,(15.43)with
A=A 0(
1 −
T∗
T
)
, A 0 ,C> 0 ,
2
9
B^2 <A 0 C.
This ansatz is motivated as follows. The specific entropy is the sum ofs 0 for the
isotropic state and a contributionsaassociated with the bond orientational order. It is
assumedthatsaisgivenbysa=kmB[...],where[...]isequaltoΦLasgivenby(15.43)
but withA 0 instead ofA=A(T). Similarly, the specific volumeρ−^1 =ρ− 01 +ρa−^1
and the specific internal energyu=u 0 +uaare made up from isotropic parts and
contributions linked with the order. The standard Gibbs relation ds 0 =T−^1 (du 0 +
Pdρ 0 −^1 ), wherePis the hydrostatic pressure, then leads to
ds=T−^1 (du+Pdρ−^1 )−kB
m∂Φ
∂aμνλκ, (15.44)
with the potential defined by
−
kB
mΦ=sa−T−^1 (ua+Pρ−a^1 ). (15.45)The plausible assumption that the ordered state has a smaller energy and a smaller
specific volume, characterized byεandva, respectively, viz.ua=−^12 εaμνλκaμνλκ
andρ−a^1 =−^12 vaaμνλκaμνλκ, leads to the expression (15.43) with
A=A 0
[
1 −T−^1
m
A 0 kB(ε+Pva)]
=A 0
[
1 −
T∗
T
]
,
T∗=T∗(P)=
m
A 0 kB(ε+Pva). (15.46)