Polymers in Confined Geometry.pdf

Chapter 3
Polymers in confined geometry
In this chapter we are going to investigate the conformational behavior of polymer
chains in a confining geometry. First a short experimental motivation is given,
after which heuristic scaling relations are derived. This is followed by a more
detailed analytical calculation, starting from the limit of stiff chains in the weaklybending
rod approximation. In particular we investigate the cylindrical harmonic
potential as a generic model for confining environments with similar a symmetry.
3.1 Motivation
The interest in confinement of polymers into nano-sized structures emerges mainly
from biotechnology. As discussed in the introduction, there are various applications
encouraging our work on the problem of confinement.
One direct application is that of confinement elongation. The advantage is
that upon confining the chain stretches already in its equilibrium conformation.
There is no need of a stretching flow or elongating external force, which drives the
polymer out of its static conformation into a stretched non-equilibrium conformation.
Here we can try to understand the mechanisms leading from the unconfined
to the confined static equilibrium conformation.
An application with a large potential impact is the determination of the
contour-length L of a polymer by measuring its equilibrium apparent end-toend
distance in an confining tube of the order of the persistence length or less.
Up to now gel-electrophoresis is the tool for this sort of measurements, but it has
its problems. It is slow—a measurement takes in the order of a day—and it is
relatively imprecise. The advantage of the idea of the confinement measurement
seems therefore a faster way to obtain the data. It is a matter of minutes to
do the actual measurement and it is in principle possible to make the results
arbitrarily accurate, by just taking more pictures of equilibrium configurations.
The basic experiments are performed in nano-sized channels with a near
square geometry in the group of Bob Austin [38, 44]. They have succeeded
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