Connection points for knots: Left: In a homogeneously aligned nematic liquid crystal, the defect line surrounds the microsphere like a Saturn’s ring. Right: In a so-called chiral nematic crystal the ring is buckled like the twisted wheel of a bicycle. Image: Miha Ravnik
can now be tied systematically in the microscopic world. A team of
scientists led by Uroš Tkalec from the Jožef Stefan Institute in
Ljubljana, Slovenia, who has been working at the Max Planck Institute
for Dynamics and Self-Organization in Göttingen, Germany, since
September 2010, has now found a way to create every imaginable knot
inside a liquid crystal. Starting points of the new method are tiny
silica microspheres confined in thin liquid crystal layers. Surrounding
these microspheres, a net of fine lines is formed where the molecular
orientation of the liquid crystal is altered. The researchers discovered
a method to twist and link these lines in such a manner as to create
every knot imaginable.
are ubiquitous: We encounter them in woven materials, in the numerous
sailors’ knots, and in constantly entangled electric cables and
extension cords. When putting on their shoes, even small children learn
to master their first knots—long before they can read and write. Even
our DNA can be complicatedly knotted. From a mathematical point of view,
knots that seem completely different at first sight can belong to the
same class. The essential criterion is that one knot can be transformed
into another by means of simple deformations. The most simple example is
a rubber band. Topologically speaking, every shape you can create from
it without cutting open the loop and joining it back together is
equivalent to the initial rubber band. A completely different knot, for
example, is the trefoil knot. This knot cannot easily be tied from an
intact rubber band. Furthermore, several interlocked loops can
constitute even more complex structures.
this tidy mathematical system organizing the general jumble of knots,
one question remains: Can every conceivable knot be implemented in a
microscopic, physical system? In his most recent study Uroš Tkalec found
such a system, in which complex knots can be created systematically:
silica microspheres within an only slightly thicker nematic liquid
crystal film confined between two glass plates. Such liquid crystals
also pose the basis of common LCD displays.
glass plates were treated in such a manner as to force the liquid
crystalline molecules to align parallel to the surface”, explains
Tkalec. A single silica microsphere entering the layer changes the
surrounding alignment substantially: around the sphere a ring-shaped
region forms in which no preferred direction can be discerned.
Scientists refer to such disruptions in the molecular order as defect
lines. Since the defect ring surrounding a microsphere reflects light
differently than the rest of the liquid crystal, it can be easily
looks as if every microsphere were surrounded by its own ring–similar
to the planet Saturn”, explains Tkalec (see figure 2, left). These
Saturn’s rings are oriented perpendicularly to the average orientation
of molecules between the glass plates. If several microspheres are
confined to such thin nematic layers, they can be moved together and
arranged in lines by using a laser, much like using a pair of tweezers.
The rings then join to form more complex, entangled lines surrounding
the aligned spheres.
A simple loop cannot easily be transformed into a trefoil knot. It would be necessary to cut it open and then paste it back to together. Image: Max Planck Institute for Dynamics and Self-Organization
in such rows of spheres no knots can be assembled”, says Tkalec.
Creating knots requires the defect rings of neighbouring microspheres to
be able to attach to each other in two directions. In order to achieve
this, the scientists used a “trick”: if the upper plate confining the
liquid crystal layer is turned by 90 degrees, the alignment of the
molecules is changed. While the lower molecules still point in the same
direction as before, the upper ones are also rotated by 90 degrees. In
between, the transition is gradual. Scientists refer to this as a
twisted or chiral nematic liquid crystal.
this experimental set-up, the defect rings surrounding the spheres are
slightly buckled—like a buckled wheel of a bicycle”, says Tkalec. The
rings of neighbouring spheres can therefore cross and link: a crucial
requirement for creating knots.
an essential step, the researchers discovered a way of manipulating the
regions between the spheres by joining and separating neighbouring
rings. First, they heated the region between the spheres with a laser.
This destroys the characteristic alignment of the molecules. After
switching off the laser, the alignment is re-established—but often in a
different way than before. Thus, it is possible to join rings that
bypassed each other before or reconnect rings in a different way.
the researchers not only proved sleight of hand in the experimental
handling of microspheres and lasers. In the theoretical part of their
study they showed that for every conceivable knot a mathematically
equivalent knot can be found which can be implemented in this way.
the help of microspheres in a chiral nematic liquid crystal, we can
create practically every knot that you can imagine”, says Tkalec.
The researchers now hope that these findings will help to better understand the complex knotting of DNA.
knotting of DNA molecules, for example, plays an important role in many
vital processes such as replication or transcription of DNA”, says Uroš
addition, the strategy may boost the assembly of reconfigurable optical
circuits in soft materials which would guide a light in future photonic