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Shnerk's first surface and the geometry of materials
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August 9, 2006
C.D. Santangelo and R.D. Kamien, "Elliptic Phases: A Study of the Nonlinear Elasticity of Twist-Grain Boundaries,"
Phys. Rev. Lett. 96 (2006) 137801.
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An analytical model of Shnerk's first surface, given
by a direct sum of parallel, screw dislocations.
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Abstract:
We develop an explicit and tractable representation
of a twist-grain-boundary phase of a smectic A liquid crystal.
This allows us to calculate the interaction energy between grain
boundaries and the relative contributions from the bending and
compression deformations. We discuss the special stability of
the 90 degree grain boundaries and discuss the relation of this
structure to the Schwarz D surface.
Smectic liquid crystals are layered materials. One the one hand, the
layers resist bending. On the other, they prefer to be equally
spaced. These two features are in conflict and the frustration between
them leads to structures with complex topologies (for e.g.,see the
figure). Resolving the conflict between layer spacing and layer bending is
an essentially three-dimensional geometry problem. The simplest
smectic is the smectic-A in which the molecules making up the layers align
with normal vector to the layers.
The twist-grain boundary phase of smectic liquid crystals is an example of
exactly this type of geometric frustration. Imagine trying to make a chiral
smectic-A: the molecules want to twist around each other, yet they want to
align along the normal to the layers. These two conditions cannot be
simultaneously, yet in some materials they can be partially satisfied by
allowing the presence of defects. Away from the core of the defects we have
smectic-A order; inside the core the smectic melts but the molecules can
twist. The result is a phase in which there exists an ordered configuration
of defects, as shown in the figure. Making models for such structures
can be quite challenging because of the highly complex topology: notice that
the layers twist as you move along the defects. You can think of this structure as
being a set of equally-spaced layers that rotate by ninety degrees along any plane of
defects.
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To understand these structures, it is useful to treat the defects as being the primary degrees of freedom
of the system. However, in many materials the defects are close together and strongly interacting.
In this paper, we've taken a step toward constructing the set of surfaces by "summing" defects together
directly in an analytical, closed form. From this construction, we can calculate the energy and stability
of the resulting phase even in the strongly interacting limit. The catch is, our construction only approximates
the true shape of the layers. Amazingly, the surface we have constructing has the same topology as
an old and well-known surface called the Schwarz D surface.
Equally amazing: we could have constructed exactly the same surface by summing defects in any one of three
perpendicular directions. In fact, this triality (if there were two ways of doing it, it would be a duality) is crucial
to our construction. These exotic, hidden defects can be exploited to make even more complicated variations
of the twist-grain boundary phase that, nevertheless, occur experimentally.
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Last updated August 24, 2006
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