30
3 Broken Symmetry
usually due to some rigid elements, like double bonds and benzene rings. The nematic alignment is characterized by a director, which is like a vector without an
arrow, as shown by the spindles in Fig. 3.7. Circular symmetry around the director may be lost; then the nematic becomes biaxial. Nematic order may emerge not
only on a molecular scale, but also on a mesoscopic scale. Cells can be polarized
in anisotropic tissues (Sect. 6.4) and oblong bacteria also tend to align nematically
(Sect. 7.3).
The sketches in Fig. 3.7 show other ordered liquid phases. In the cholesteric
phase, the prevailing orientation rotates around some axis. In smectics, molecules
are layered; within each layer, they are oriented normally in the A phase, at a certain
angle in the C phase, at at an angle rotated between one layer and the next in the C ∗
phase; within each layer, their positions are disordered. As in all phase transitions,
order diminishes as temperature grows, so that, for example, a solid may melt into
a smectic, which melts into nematic, and then to an isotropic liquid.
Liquid crystals possess a kind of elasticity which, unlike in solids, does not hold
back deformations but only works to orient them uniformly. In nematics, the change
of orientation can be of three kinds: splay, bend, and twist, as sketched in the three
left-hand panels of Fig. 3.8. Since there is no translational order, nematics can flow
like normal fluids, only their viscosity is anisotropic. Smectics are already sensitive
to the distortion of the shape of their layers and are only mobile within the layers.
If an isotropic liquid is “frozen” into the nematic state, the orientation will be
different at different locations, and there will be a lot of defects – points where the
orientation becomes indefinite. Defects are not just arbitrary flaws: they obey precise
laws of topology (Kleman, 1983). The topological charge of a defect is measured
by the rotation of the director along a surrounding contour. Stable defects with the
lowest energy have the lowest possible charge. Since the director is invariant under rotation by 180 ◦ , defects may have charge one-half, either positive or negative.
Paired defects of this kind in the plane are shown in Fig. 3.8. This picture can also be
viewed as a cross-section of a half-charged line defect in three dimensions. If such
a line forms a closed contour, it can collapse into a point defect, called, for obvious
reasons, a hedgehog (see the rightmost panel of Fig. 3.8). Defects of opposite signs
attract each other. If the pair of defects shown in Fig. 3.8 merges and annihilates,
the alignment will become uniform, at least locally, but it is impossible to eliminate
all defects if the nematic orientation is required to remain at a certain angle to the
boundary or on a closed surface, like a sphere.
From a practical point of view, what is special in liquid crystals is their optical
properties. Due to their anisotropy, liquid crystals polarize light. Light, as a transverse wave, has two polarization directions, and a liquid crystal transmits the one
aligned with its director; it also rotates the polarization of light when oriented at
some angle to a light ray. The orientation of the director is easily controlled by
treating the confining surfaces and using electric or magnetic fields. In the pixels of
standard displays, the light beam passes through a polarizer that polarizes it parallel
to the orientation of the liquid crystal, set by treating its surface. The orientation of
both the polarizer and the liquid crystal at the other end are turned through 90 ◦ , and
light, rotated by the twisted liquid crystal, passes through. When a voltage is turned
3 Broken Symmetry
usually due to some rigid elements, like double bonds and benzene rings. The nematic alignment is characterized by a director, which is like a vector without an
arrow, as shown by the spindles in Fig. 3.7. Circular symmetry around the director may be lost; then the nematic becomes biaxial. Nematic order may emerge not
only on a molecular scale, but also on a mesoscopic scale. Cells can be polarized
in anisotropic tissues (Sect. 6.4) and oblong bacteria also tend to align nematically
(Sect. 7.3).
The sketches in Fig. 3.7 show other ordered liquid phases. In the cholesteric
phase, the prevailing orientation rotates around some axis. In smectics, molecules
are layered; within each layer, they are oriented normally in the A phase, at a certain
angle in the C phase, at at an angle rotated between one layer and the next in the C ∗
phase; within each layer, their positions are disordered. As in all phase transitions,
order diminishes as temperature grows, so that, for example, a solid may melt into
a smectic, which melts into nematic, and then to an isotropic liquid.
Liquid crystals possess a kind of elasticity which, unlike in solids, does not hold
back deformations but only works to orient them uniformly. In nematics, the change
of orientation can be of three kinds: splay, bend, and twist, as sketched in the three
left-hand panels of Fig. 3.8. Since there is no translational order, nematics can flow
like normal fluids, only their viscosity is anisotropic. Smectics are already sensitive
to the distortion of the shape of their layers and are only mobile within the layers.
If an isotropic liquid is “frozen” into the nematic state, the orientation will be
different at different locations, and there will be a lot of defects – points where the
orientation becomes indefinite. Defects are not just arbitrary flaws: they obey precise
laws of topology (Kleman, 1983). The topological charge of a defect is measured
by the rotation of the director along a surrounding contour. Stable defects with the
lowest energy have the lowest possible charge. Since the director is invariant under rotation by 180 ◦ , defects may have charge one-half, either positive or negative.
Paired defects of this kind in the plane are shown in Fig. 3.8. This picture can also be
viewed as a cross-section of a half-charged line defect in three dimensions. If such
a line forms a closed contour, it can collapse into a point defect, called, for obvious
reasons, a hedgehog (see the rightmost panel of Fig. 3.8). Defects of opposite signs
attract each other. If the pair of defects shown in Fig. 3.8 merges and annihilates,
the alignment will become uniform, at least locally, but it is impossible to eliminate
all defects if the nematic orientation is required to remain at a certain angle to the
boundary or on a closed surface, like a sphere.
From a practical point of view, what is special in liquid crystals is their optical
properties. Due to their anisotropy, liquid crystals polarize light. Light, as a transverse wave, has two polarization directions, and a liquid crystal transmits the one
aligned with its director; it also rotates the polarization of light when oriented at
some angle to a light ray. The orientation of the director is easily controlled by
treating the confining surfaces and using electric or magnetic fields. In the pixels of
standard displays, the light beam passes through a polarizer that polarizes it parallel
to the orientation of the liquid crystal, set by treating its surface. The orientation of
both the polarizer and the liquid crystal at the other end are turned through 90 ◦ , and
light, rotated by the twisted liquid crystal, passes through. When a voltage is turned