Polymers in Confined Geometry.pdf

58 CHAPTER 5. SIMULATION RESULTS
R 2
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
ɛ
6
7
exact
simulation
Figure 5.1: Comparison of the simulation for a polymer in bulk solution c = 0 to the exact
formula eq. (2.21). The dependence of R 2 on the flexibility ɛ is shown.
5.1 The unconfined chain
Before one can start using a simulation algorithm to produce data for a situation
where no analytical comparison is available, it should be validated by comparing
the results with well known limiting cases where analytical results are available.
This is the case for a polymer in bulk solution, where the dependence of the
second moment of the end-to-end vector distribution on the persistence length is
well known (cf. eq. (2.21)). This limit can be simulated using our algorithm for
a collision parameter c = 0. The results in figure 5.1 show that there is perfect
agreement within the statistical error.
An additional check can be done with the tangent-tangent correlation function
which should show simple exponential behavior. In the logarithmic plot 5.2 this
results in a straight line. The simulation data for e.g. lp = 10 again fits eq. (2.12)
perfectly within the errorbars.
In addition, the radial distribution function has also been checked against
known results, such that the algorithm—at least concerning the bending energy—
is fully validated.
5.2 The harmonic potential
A convenient starting point for the discussion of confinement on the conformation
of a polymer is a harmonic potential with cylindrical symmetry. This has the
advantage that we can compare with analytical results at least in the stiff limit
(see section 3.3).
8
9
10