Relaxation of a Single Knotted Ring Polymer
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)Abstract
The relaxation of a single knotted ring polymer
%with the trivial or the trefoil knot
is studied by Brownian dynamics simulations.
The relaxation rate $łambda_q$ for the wave number $q$ is estimated
by the least square fit of
the equilibrium time-displaced correlation function
$C_q(t) = N^-1 \sum_i \sum_j C_i,j(t)
2\piq(j-i)/N $
to a double exponential decay at long times.
Here,
$N$ is the number of segments of a ring polymer and
$C_i,j(t)$ denotes the equilibrium time-displaced correlation function
of the positions of the $i$th and the $j$th segments
relative to the center of mass of the polymer.
\par
Figure 1 shows log-log plots of $łambda_q$ versus $q/N$
for the single ring polymer with
(a) the trivial knot and (b) the trefoil knot.
The solid symbols represent the relaxation rates for $q=1$ and
the open symbols represent those for $q>1$.
The relaxation rate distribution
of a single ring polymer with the trivial or the trefoil knot
appears to behave as
$ łambda_q A(1/N)^x $ for $q=1$ and
$ łambda_q A'(q/N)^x' $ for $q>1$.
In the case of the trivial knot,
$x x' 2.15$ and $ A < A'$.
These exponents are similar to that found for a linear polymer chain.
The topological effect appears as the difference between
the amplitudes, which does not appear for a linear polymer chain.
In the case of the trefoil knot,
$x 2.52$, $x' 1.95$ and $A > A'$.
It is found that
the slowest relaxation rate for each $N$ is given by $łambda_q$
with $q=2$ for the small values of $N$ and
that with $q=1$ for the large values of $N$.
This transition is considered to be caused by
the change of the structure of the ring polymer
from a ``uniform'' state for small $N$,
where the knotted part is extended widely along the ring polymer,
to a ``phase segregated'' state for large $N$,
where the knotted part is localized to a part of the ring polymer
and the rest of the ring polymer behaves like
a ring polymer with the trivial knot.
See figure 2.
\par
This localization of the knotted part is
confirmed by the analysis of the ``average structure.''
The ``average structure'' is obtained self-consistently
by averaging many structures obtained from simulations,
which are translated and rotated
to be matched to the ``average structure.''
Note that
the ``average structure'' does not necessarily
preserve the topology of the original structures.
The ``average structures'' of the ring polymer with the trefoil knot
are shown in figure 3 for $N=24$ and $192$.
The ``average structure''
changes from the double-loop structure for small $N$
to the single-loop structure for large $N$.
This change corresponds to the localization of the knotted part
of the ring polymer.