2.2 Landau’s Phenomenology
47
2.2.6 Implication and Outcome
Previous Sects. 2.2.3 and 2.2.4 revealed that the different nature of phase transition
is related to the difference in the system symmetry. This result means that a detailed
inspection over the system far from the transition point tells the nature of the transition. A remarkable example of this kind is a transition between isotropic liquid and
nematic liquid crystal with uniaxial order [16]. Since the usual nematic liquid crystal is not polar, individual molecules can be regarded as rods symmetric concerning
head and tail. This headless nature denies using the average of the orientation vector s
because it can vanish even if all molecules align perfectly along an axis. Alternately, a
proper order parameter for the nematic liquid crystal is given by φ = (3 cos θ
2
− 1)/2
with θ being the angle from the uniaxial axis. This nematic order parameter indeed
vanishes for the perfect disorder. It is interesting to see that the perfect alignment
within a plane perpendicular to the nematic axis (θ = π/2) accompanies φ = −
1
2
.
This fact indicates that the different signs of the nematic order parameter correspond
to entirely different states. Thus, the expansion of the thermodynamic potential of
the isotropic liquid in terms of the nematic order parameter unavoidably has a cubic
(third-order) term. The analysis in Sect. 2.2.4 indicates that the transition should be
discontinuous. Similar discussions are possible on the nature of phase transitions
based on the symmetry of the disordered phase (with vanishing magnitude of the
relevant order parameter).
The restriction on the functional form of the thermodynamic potential of the disordered phase in terms of order parameters exerts some conditions for the possibility
of continuous transition. Since representations of the space group, which characterizes the symmetry of crystals, depends on the wavevector of modulation wave, the
order parameter accompanies the wavevector. For phase transitions between crystalline phases, conditions for continuous phase transition appear as selection rules
for allowed wavevectors [2, 3, 17]. It is interesting to see that a general wavevector
that drives a transition to an incommensurately modulated phase (incommensurate
phase) is mostly compatible with a continuous transition while rational wavevectors leading to a simple multiplication of unit cell is often prohibited from driving a
continuous one.
It is often the case that plural order parameters, which are physically different
from each other, are involved in a phase transition. In such cases, the argument
within Landau’s phenomenology offers the restriction on the allowed form of the
coupling of order parameters. Imagine a phase transition from the paraelectric phase
to the ferroelectric phase with spontaneous polarization p. Since the ferroelectricity
usually accompanies atomic (or molecular) displacement(s) in the crystal, the elastic
deformation u is mostly induced upon the appearance of p. For such systems, the
simplest form of thermodynamic potential contains a coupling term proportional
to up
2 because the resultant deformation is independent of the sign of p. Another
example is found with a detailed analysis in [18].
The coupling term of the last-mentioned form, up
2 , deserves another remark.
To this point, we have implicitly assumed a single instability. Namely, we have
47
2.2.6 Implication and Outcome
Previous Sects. 2.2.3 and 2.2.4 revealed that the different nature of phase transition
is related to the difference in the system symmetry. This result means that a detailed
inspection over the system far from the transition point tells the nature of the transition. A remarkable example of this kind is a transition between isotropic liquid and
nematic liquid crystal with uniaxial order [16]. Since the usual nematic liquid crystal is not polar, individual molecules can be regarded as rods symmetric concerning
head and tail. This headless nature denies using the average of the orientation vector s
because it can vanish even if all molecules align perfectly along an axis. Alternately, a
proper order parameter for the nematic liquid crystal is given by φ = (3 cos θ
2
− 1)/2
with θ being the angle from the uniaxial axis. This nematic order parameter indeed
vanishes for the perfect disorder. It is interesting to see that the perfect alignment
within a plane perpendicular to the nematic axis (θ = π/2) accompanies φ = −
1
2
.
This fact indicates that the different signs of the nematic order parameter correspond
to entirely different states. Thus, the expansion of the thermodynamic potential of
the isotropic liquid in terms of the nematic order parameter unavoidably has a cubic
(third-order) term. The analysis in Sect. 2.2.4 indicates that the transition should be
discontinuous. Similar discussions are possible on the nature of phase transitions
based on the symmetry of the disordered phase (with vanishing magnitude of the
relevant order parameter).
The restriction on the functional form of the thermodynamic potential of the disordered phase in terms of order parameters exerts some conditions for the possibility
of continuous transition. Since representations of the space group, which characterizes the symmetry of crystals, depends on the wavevector of modulation wave, the
order parameter accompanies the wavevector. For phase transitions between crystalline phases, conditions for continuous phase transition appear as selection rules
for allowed wavevectors [2, 3, 17]. It is interesting to see that a general wavevector
that drives a transition to an incommensurately modulated phase (incommensurate
phase) is mostly compatible with a continuous transition while rational wavevectors leading to a simple multiplication of unit cell is often prohibited from driving a
continuous one.
It is often the case that plural order parameters, which are physically different
from each other, are involved in a phase transition. In such cases, the argument
within Landau’s phenomenology offers the restriction on the allowed form of the
coupling of order parameters. Imagine a phase transition from the paraelectric phase
to the ferroelectric phase with spontaneous polarization p. Since the ferroelectricity
usually accompanies atomic (or molecular) displacement(s) in the crystal, the elastic
deformation u is mostly induced upon the appearance of p. For such systems, the
simplest form of thermodynamic potential contains a coupling term proportional
to up
2 because the resultant deformation is independent of the sign of p. Another
example is found with a detailed analysis in [18].
The coupling term of the last-mentioned form, up
2 , deserves another remark.
To this point, we have implicitly assumed a single instability. Namely, we have