Polymers in Confined Geometry.pdf

5.2. THE HARMONIC POTENTIAL 63
behavior of the tangent-tangent correlation function. The actual polymer in the
simulation has free ends and is therefore much less constrained in comparison to
the analytical calculation where the fluctuation of the boundaries are suppressed.
The influence of the free ends was already seen on the undulations in figure 5.3
as a large increase of the deviation from the analytical result. But also in figure
5.4(c) a reminiscence of anticorrelations of the segments can be seen as a small
shoulder at |s − 0.5| ≈ 1/c = 1/3.
Figure 5.4(b) shows, for a collision parameter c = 5 · 10 −4 , an effectively free
polymer. The analytical result fits the data perfectly over the whole polymer
length, except for a very small deviation at the ends. This behavior is expected
since the analytical calculation is performed in the weakly-bending rod limit,
which gives the correct result for unconfined stiff polymers up to first order in ɛ,
as shown in the discussion leading to eq. (3.36). Therefore the small deviation
are only due to the limits of the weakly-bending rod approximation (terms of
order ɛ 2 ) and not due to boundary effects.
Finite-size effects have only a very small influence on the results obtained for
the tangent-tangent correlation. The overall shape is not affected by increasing
the number of segments N. Only small global shifts of the whole correlation
function can be observed. This has only an effect on integral quantities as the
end-to-end distance, which will be discussed in the following section.
After these preliminary investigations towards the validity of the dependence
of the observables on the scaling parameter c/ɛ, we now turn to the systematic
investigation using a scaling plot.
5.2.3 Scaling plot
We have already discussed the influence of confinement on the undulations and
the tangent-tangent correlation. Now we will turn to quantities, which are of
particular interest in experiments, where it is easier to measure the end-to-end
distance and related quantities than the very small undulation or a correlation
function of the tangents.
We want to present the data in a compact form, which works out the scaling
properties expected by our investigation up to this point. In a suitable plot
we expect a data collapse. Looking back at eqs. (3.12) and (3.38) we see that
plotting the mean-square end-to-end distance 〈R 2 〉 versus c/ɛ is expected to show
the data collapse. But before turning to the plot we first give an overview over
the various parameter regimes and than try to find out in which range the scaling
is expected to start, since it is not universal for all parameters as noted in the
preceding section.