Polymers in Confined Geometry.pdf

70 CHAPTER 5. SIMULATION RESULTS
R
R ||
Figure 5.10: Behavior of the end-to-end distances of flexible chains upon confinement.
A representative example of these observables for a flexible chain ɛ = 10 is
shown in figure 5.9, where the mean-square and mean-square parallel end-to-end
distance, 〈R2 〉 and R2
|| , are plotted as a function of the collision parameter c.
The full end-to-end distance exhibits a ‘dip’ (local minimum) for c ≈ 1, when the
chain begins to notice the tube through deflections. This dip is absent for R2
|| .
The two limits, c → 0 and c → ∞, are easily understood: In the limit of
an unconfined chain (c < 1) the average of the z-component is exactly a third
of the full average R2
2
|| = 〈R 〉 /3. In the opposite limit, when the confinement
becomes larger (c > 20), both observables, 〈R2 〉 and R2
|| , behave identically
since the polymer is oriented mainly along the tube. Therefore the end-to-end
distance is nearly parallel to the tube axis.
The dip—which has also been observed without explanation in [43]—can be
explained as illustrated in figure 5.10. Upon confinement of the flexible chain,
the initially randomly oriented end-to-end distance is hindered in the perpendicular
directions. This implies that first the end-to-end distance is changed
predominantly perpendicular to the tube axis, only decreasing the value of 〈R2 〉
but leaving R2
|| unchanged. Formation of the dip should start, as seen in figure
5.9, at a value of c 1. Upon further confinement the end-to-end distance
slowly starts giving way in the direction along the tube, finally also increasing
the parallel component.
The behavior of the end-to-end distances upon confinement can also be seen
in the shape of the radial distribution functions (see next section 5.2.5).
The dip is present in all end-to-end distances introduced in section 4.6, as it
is absent for all parallel end-to-end distances.
5.2.5 Radial distribution function
In this section we are going to investigate the probability distribution of the
end-to-end distance or radial distribution function (RDF) (see section 3.3.3 for
a discussion of the RDF in the unconfined case). Figures 5.11 show some representative
numerical results for a chain with flexibility ɛ = 10 and for increasing
collision parameter c.
The RDF for the end-to-end distance R is shown in figure 5.11(a) and for
δR ||
δR
d