Polymers in Confined Geometry.pdf

66 CHAPTER 5. SIMULATION RESULTS
R 2
1
0.8
0.6
0.4
0.2
0
0.1
1
10
c/ɛ
100
ɛ = 0.1
ɛ = 0.5
ɛ = 1
ɛ = 5
ɛ = 10
ɛ = 50
ɛ = 300
Figure 5.6: Scaling plot for the mean-square end-to-end distance R 2 versus the scaling
parameter c/ɛ for various flexibilities ɛ from the stiff to the flexible polymer limit.
square end-to-end distance 〈R 2 〉 as function of the scaling parameter c/ɛ is shown
for the whole range of flexibilities ɛ, from the stiff up to the flexible limit. It can
be seen that a universal behavior is visible to start around c/ɛ 5. Taking a
lower envelope of the data reveals the expected scaling.
The deviations from a lower envelope in this range can be explained by finitesize
effects of the simulated chains. Already for ɛ ≈ 5 differences from the analytic
result eq. (3.38) can be observed. This is explicitly shown in figure 5.7 for the
mean-square end-to-end distance 〈R 2 〉 versus the scaling parameter c/ɛ for a
flexibility ɛ = 5. The simulation data is obtained by increasing the number N of
the segments up to 1000. Since a simulation for chains of length N = 1000 takes
in the order of a week per data point, only single points have been simulated.
The inset gives a zoomed view of the area where the major effect of finite-size is
seen. The data obtained for N = 1000 is seen to fit the analytic result perfectly.
The data converges, in this range, for decreasing segments size to the analytic
result.
These finite-size deviations increase for more flexible chains: The condition
that the segment length l = 1/N should be at least about the size of the persistence
length or smaller—otherwise the persistence length is artificially increased
due to finite-size effects—forces us to use more segments for flexible chains.
Therefore deviations due to finite-size are increasing even more. But using much
more segments than N = 1000 is problematic due to the simulation time needed.
The form of the mean-square end-to-end distance for very flexible chain is
given by the limit of the ideal chain without persistence, the freely jointed chain
(FJC). For c/ɛ 1 the function stays effective constant and for c/ɛ 1 the
function behaves like the approximation for strong confinement eq. (3.9), only
1000