32
3 Broken Symmetry
(1975). When this material is “actuated”, prompting a phase transition from the
isotropic to the nematic state, it elongates along the director and accordingly shrinks
in the normal directions to preserve its volume, as shown in the two lower left-hand
panels of Fig. 3.9. It goes the opposite way when the transition is reversed. Elongation is stronger for main-chain elastomers. In this way, a liquid crystal elastomer can
really work as a muscle, lifting a considerable weight, but much more can be accomplished through reversible phase transitions under the action of light, temperature,
or chemical agents.
In the example shown in the upper right-hand panel of Fig. 3.9 (Palagi et al,
2016), a wave enforcing a transition from the disordered to the ordered state and
back makes this nemato-elastic “worm” crawl in the direction shown by the arrow.
If the phase transition is enforced on one side only, as in a layered material or when
only one side of a flat strip is heated or illuminated, the strip bends, as shown in
the lower right-hand panel of the same figure (Camacho-Lopez et al, 2004). Strings,
sheets, or shells of liquid crystal elastomers can be made to walk and swim, and
even used to construct soft robots – more on this in Sect. 8.2.
3.4 Non-Equilibrium Patterns
Symmetry breaking takes place on a macroscopic scale in systems sustained far
from equilibrium by external fluxes. Broken symmetry is most evident in fluid
mechanics. Water and air are never still, and neither are fluid-like granular media
forming sand ripples and dunes. The Earth itself, fluid on a geological time scale,
breaks its spherical symmetry by its ever changing rugged relief. While we enjoy
varied landscapes and the play of waves, we have to distinguish between symmetry broken by outside interference, when waves are caused either by wind or local
perturbations, like throwing a pebble, and spontaneous symmetry breaking – selforganization, which can be inferred in a pattern of cloudlets we often see when
looking up to the skies or down from an airplane window. A hurricane is a powerful
solitary structure spontaneously emerging from humidity above warm ocean waters
and winds and currents driven by the rotating Earth.
The basic scheme of self-organization of this kind, repeated not only in hydrodynamics but in various non-equilibrium processes, is shown in Fig. 3.10 (left panel).
We start with a featureless plane. The external input is directed perpendicularly,
without disturbing the symmetry – but the result is a pattern. Since the vertical direction is reserved for the external flux sustaining the system in a non-equlibrium
state, the emerging patterns are two-dimensional. The most common pattern is a
hexagonal grid, generated by the resonant interaction of three waves forming a regular (equilateral) triangle. It can be distorted, with differently oriented patches separated by strings of defects, as in the central panel of Fig. 3.10. Striped patterns can
be seen as well, usually distorted to a labyrinthine tangle, sometimes coexisting with
hexagons, as in the right-hand panel of Fig. 3.10.
3 Broken Symmetry
(1975). When this material is “actuated”, prompting a phase transition from the
isotropic to the nematic state, it elongates along the director and accordingly shrinks
in the normal directions to preserve its volume, as shown in the two lower left-hand
panels of Fig. 3.9. It goes the opposite way when the transition is reversed. Elongation is stronger for main-chain elastomers. In this way, a liquid crystal elastomer can
really work as a muscle, lifting a considerable weight, but much more can be accomplished through reversible phase transitions under the action of light, temperature,
or chemical agents.
In the example shown in the upper right-hand panel of Fig. 3.9 (Palagi et al,
2016), a wave enforcing a transition from the disordered to the ordered state and
back makes this nemato-elastic “worm” crawl in the direction shown by the arrow.
If the phase transition is enforced on one side only, as in a layered material or when
only one side of a flat strip is heated or illuminated, the strip bends, as shown in
the lower right-hand panel of the same figure (Camacho-Lopez et al, 2004). Strings,
sheets, or shells of liquid crystal elastomers can be made to walk and swim, and
even used to construct soft robots – more on this in Sect. 8.2.
3.4 Non-Equilibrium Patterns
Symmetry breaking takes place on a macroscopic scale in systems sustained far
from equilibrium by external fluxes. Broken symmetry is most evident in fluid
mechanics. Water and air are never still, and neither are fluid-like granular media
forming sand ripples and dunes. The Earth itself, fluid on a geological time scale,
breaks its spherical symmetry by its ever changing rugged relief. While we enjoy
varied landscapes and the play of waves, we have to distinguish between symmetry broken by outside interference, when waves are caused either by wind or local
perturbations, like throwing a pebble, and spontaneous symmetry breaking – selforganization, which can be inferred in a pattern of cloudlets we often see when
looking up to the skies or down from an airplane window. A hurricane is a powerful
solitary structure spontaneously emerging from humidity above warm ocean waters
and winds and currents driven by the rotating Earth.
The basic scheme of self-organization of this kind, repeated not only in hydrodynamics but in various non-equilibrium processes, is shown in Fig. 3.10 (left panel).
We start with a featureless plane. The external input is directed perpendicularly,
without disturbing the symmetry – but the result is a pattern. Since the vertical direction is reserved for the external flux sustaining the system in a non-equlibrium
state, the emerging patterns are two-dimensional. The most common pattern is a
hexagonal grid, generated by the resonant interaction of three waves forming a regular (equilateral) triangle. It can be distorted, with differently oriented patches separated by strings of defects, as in the central panel of Fig. 3.10. Striped patterns can
be seen as well, usually distorted to a labyrinthine tangle, sometimes coexisting with
hexagons, as in the right-hand panel of Fig. 3.10.