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\title{Granular Chains}
\author{E.~Ben-Naim, Z.~A.~Daya, and R.~E.~Ecke}
\affiliation{Theoretical Division, Center for Nonlinear Studies,
Condensed Matter \& Thermal Physics Group, Los Alamos National
Laboratory, Los Alamos, NM 87545}
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Granular chains are made of hollow steel balls connected by thin metal
rods. They can be purchased at your neighborhood hardware store. Add a
shaker and a little bit of imagination and you have a rich laboratory
for polymer physics \cite{bzve}, nonequilibrium statistical mechanics
\cite{hdbe}, nonlinear dynamics \cite{edrb,bsew}, and biological
physics \cite{bo}. The attached video and the brief summaries below
highlight a series of our recent experiments in vibrated granular
chains.

Like a tie on a shoestring, a knot in a granular chain comes undone
upon vibration. The time it takes for the knot to untie grows
quadratically with the number of balls in the chain, indicative of a
diffusive motion. Moreover, the shape of the distribution of opening
times is universal, as it is independent of the length of the
chain. The statistics of the unraveling times are governed by the
number of crossing points. For example, a simple trefoil knot has
three crossing points (top figure). The diffusive motion of the
crossing points governs the knot dynamics \cite{bzve}.

The chains can be easily made into rings and by controlling the
vibration strength, it is possible to control the chain topology. For
example, a gently vibrated figure-8 remains forever in that
configuration (bottom left figure). We observed that as the figure-8
explores various configurations, it is much more likely for one of the
two loops to be small while the other is large. This is the first
observation of the entropic tightening phenomenon that is expected
from the equilibrium statistical mechanics of macromolecules. Although
the chain is not in equilibrium, being driven in an athermal fashion,
the structure of the configuration phase space overwhelmingly favors
the small-loop large-loop state \cite{hdbe}.

We also observed that in a narrow range of driving parameters, a
linear chain spontaneously coils into a stable spiral (bottom right
figure). In this state the chain undergoes a slow rotation, keeping
itself wound while the core of the spiral undergoes a ratcheting
motion.  Shaking excites vibrational motion with dominant transverse
oscillations.  Small asymmetries between the two ends of the chain are
responsible for this behavior: the spiral is stable only when one of
the two end balls is its the core \cite{edrb} .


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\begin{thebibliography}{99}

\bibitem{bzve} E.~Ben-Naim, Z.A.~Daya, P.~Vorobieff, and R.E.~Ecke,
 Phys. Rev. Lett., {\bf 86}, 1414 (2001).

\bibitem{hdbe} M.B.~Hastings, Z.A.~Daya, E.~Ben-Naim, and
 R.E.~Ecke, Phys. Rev. E., {\bf 66}, 025102(R) (2002).

\bibitem{edrb}
R.~E.~Ecke, Z.~A.~Daya, M.~K.~Rivera, and E.~Ben-Naim, 
MRS Symposium Proceedings {\bf 759}, 129 (2003).

\bibitem{bsew}
A.~Belmonte, M.~J.~Shelley, S.~T.~Eldakar, and C.~H.~Wiggins, 
Phys. Rev. Lett. {\bf 87}, 114301 (2001). 

\bibitem{bo}
J.~S.~Olafsen and B.~Bammes,
Chaos {\bf 14}, S9 (2004).

\end{thebibliography}
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