Polymers in Confined Geometry.pdf

52 CHAPTER 4. SIMULATION METHODS
detail will be the cylindrical symmetric harmonic potential as already introduced
in eq. (3.20). The accuracy of this approximation and how to relate it to the
other potentials will be discussed in section 5.
4.5.1 Harmonic potential
The configuration energy function ∆e corresponding to eq. (4.25) is given by the
discretized version of eq. (3.20)
N−1
∆e = βH ({ti}) = −k ˆti · ˆti+1 + g
2
i=1
N 2
xi + y 2 i , (4.29)
such that we now need the tangents and the positions of the beads in the simulation.
The latter can be directly calculated from the tangents, but storing them
is faster.
We have to relate the prefactor g to the Odijk deflection length ld by discretizing
the harmonic potential part of eq. (3.20). Additionally one has to use
dimensionless units in the simulation. Hence all positions are measured in units
of the segment length l
γ
2
L
0
ds x(s) 2 + y(s) 2 = γl3
2
→ γl3
2
g/2β
Nl
0
ds
l
i=0
x(s)
l
2
2
y(s)
+
l
N 2
xi + y 2 i , (4.30)
where xi, yi are normalized by the segment length l = L/N. Upon using the
one finds
definition ld := (4κ/γ) 1
4 ⇒ γ = 4κ/l 4 d
g = βγl 3 = 4βκl3
l 4 d
i=0
(2.20)
= 4lpl 3
l 4 d
= 4c4
N 3 . (4.31)
ɛ
The actual parameter used for the simulations is the collision parameter c as
defined in eq. (3.3).
With these prerequisites the simulation can be implemented straightforwardly.
Only one additional MC move must be added since the space is not isotropic
anymore. Now global movements of the polymer perpendicular to the tube axis
must be attempted. This can be implemented by using a virtual segment ‘zero’.
If this is randomly selected, all positions are shifted by a random amount out of
±∆r⊥. The detailed balance condition is therefore obeyed. This maximum shift
is, along with the maximal rotation angle δα, adjusted in the initial simulation
phase. The z-position of the first bead can be kept fixed, since there is still the
rotational symmetry around the tube/z-axis.