Polymers in Confined Geometry.pdf

72 CHAPTER 5. SIMULATION RESULTS
the parallel end-to-end distance R|| in figure 5.11(b). For an unconfined chain
(c 2), p(R) is peaked around R = 0.4 and shows Gaussian behavior, whereas
p(R||) is peaked at R|| = 0. This is simply due to a phase space factor of 4πR 2
included in p(R) resulting from sampling all directions of R. Such a phase space
factor is absent from p(R||) which samples the projection of R on the tube axis.
For intermediate values of c (6 c 20) the RDF p(R) shows an interesting
bimodal distribution. If the collision parameter c is increased from c ≈ 2 the peak
is shifted towards smaller values of R such that the distribution is significantly
skewed. Then, at about c ≈ 6 a second peak at large values of R develops, which
then grows at the expense of the peak at low values of R.
This feature is closely related to the ‘dip’ discussed in the previous section.
The initial drift of the peak towards smaller values is due to the restriction
of the polymers’ configuration perpendicular to the tube axis. Such a behavior
is expected from a linear response argument. When the confinement becomes
stronger, the end-to-end distance evades the transverse compression by
also stretching along the tube axis. As a result the RDF p(R) exhibits two
peaks, one at small and the other at large values of R. This quantifies the heuristic
arguments for the dip of 〈R 2 〉 in the previous section.
For comparison the parallel end-to-end distance distribution is shown in figure
5.11(b) for the same set of parameters. Obviously the average is growing
monotonously to larger values, not showing any bimodal behavior. We have already
eluded to the fact that p(R||) is peaked at R|| = 0 for weak confinement
(due to phase space effects). Hence, increasing the confinement can only result
in a shift of this peak to larger values.
Finally for narrow ‘tubes’ (c 30) both RDFs show exactly the same shape
since the end-to-end distance is now oriented parallel to the tube-axis.
5.3 The hard walls
In the final chapter we turn to the investigation of preliminary simulation results
concerning the conformations of polymers confined by hard walls. In particular,
we are going to analyze the scaling of the end-to-end distance similar as in section
5.2.3. We study two different cross sections of the tube, a circular and square
shaped channel, and leave other geometries for future investigations.
Since we are mainly interested in comparing with the experimental data (cf.
section 5.2.3) we concentrate on the mean apparent end-to-end distance 〈Ra〉.
The simulation data for the two different channel cross sections are shown in figures
5.12 and 5.13. In both cases one finds scaling behavior with scaling variables
(dɛ) −1 and (wɛ) −1 , respectively. Here d is the tube diameter for a circular cross
section and w the channel width for a square cross section.
Comparing both plots with each other and to figure 5.6 shows that the overall
behavior is identical and that there appears to be no major influence of the chosen