Polymers in Confined Geometry.pdf

22 CHAPTER 3. POLYMERS IN CONFINED GEOMETRY
d
ld
r
Figure 3.1: The new scales for confined polymers. The Odijk deflection length ld measures the
typical length between successive deflections along the contour. r denotes the distance between
deflection along the tube.
in confining fluorescently dyed λ-DNA strands into nano-sized channels seen in
the figures 1.1. The channels have an aspect ratio between one and two and are
of average size 440 nm down to 30 nm. These experiments provide a picture of
increasingly stretched strands when the channel becomes smaller, which is seen
for relatively long T2-DNA and the shorter λ-DNA in the figures 1.2. The value,
which can be read off from the pictures by using appropriate fitting functions, is
the apparent end-to-end distance 〈Ra〉, which is the maximal distance between
two monomers along the chain.
Here we will present first by scaling relations then by analytical approximations
on the cylindrical symmetric harmonic potential 1 —modeling the tube—
approaches to a solution of the problem of confinement and the identification
of the relevant scaling parameters. This is investigated in detail—particularly
for the relevant range of parameters for the experimentalists—by simulations in
chapter 5. The resulting “scaling plot” in section 5.2.3 can serve as a gauge for
converting measured apparent end-to-end distances into real contour lengths L.
3.2 Scaling relations
Before giving a detailed analytical calculation, it is instructive to perform a scaling
analysis starting from the relevant length and energy scales. Often it is relatively
easy to find an intuitive scaling behavior. For an unconfined polymer the stiffness
of the filament is characterized by the persistence length lp, and the thermal
energies ∼ kBT determine the typical energy scale. Upon confining a polymer to a
tube of lateral dimension d there is an additional scale, the Odijk deflection length
ld (introduced in [37]). It measures the typical distance of successive collisions of
the polymer with the walls of the confining tube. The new parameters d and ld
are shown in figure 3.1. A relation between these values can be easily established.
With r⊥ ∼ d and s ∼ ld the transverse and longitudinal length scales, respectively,
we find
kBT ∼ H ∼ (lpkBT ) d2
l 3 d
⇒ l 3 d ∼ d 2 lp , (3.1)
1 The harmonic potential being again and again the “laboratory animal” of the physicist. . .