Polymers in Confined Geometry.pdf

3.3. ANALYTICAL RESULTS 27
The mean-square average deviation from the z-axis (undulation) is thus given by
r⊥(s) 2 = x 2 (s) + y 2 (s) = 2 x 2 (s) = 2
3
s 3
lp
. (3.17)
We can again introduce the length scale ld, now following directly the line of
reasoning of Odijk in [37]. When confining the chain into a cylindrical tube of
diameter d ≪ lp, with its axis along the z-direction, the contour length ld can be
calculated by using eq. (3.17).
As a measure for the distance on the contour between two deflection points,
ld can be defined by assuming that the mean-square average lateral deviation
should be of the same order as the radius of the tube (d/2) 2
r⊥(ld) 2 ∼
2 d
2
⇒ l 3 d ∼ d 2 lp. (3.18)
In this way we reobtained the important relation eq. (3.1). By this, the scaling for
the end-to-end distance in the strongly confined regime can be obtained, stacking
linearly L/ld pieces of the polymer of length 〈z(ld) 2 〉 gives
R
L ∼ 〈z(ld) 2 〉/ld ∼ 1 − α ′
d
lp
2/3
, (3.19)
where α ′ is some unknown constant (cf. eq. (3.12)).
The results obtained here should be valid for all flexibilities as long as the
confinement is very strong. But for the intermediate regime where d ≈ R and
the polymer is semi-flexible lp ≈ L simulations become essential.
3.3 Analytical results
After this preparing scaling analysis, we address the task of analytical approximations.
In the context of analytical solutions one has to think about in what
complexity a problem is feasible to solve. Every reasonable behaving symmetric
potential is expandable in a power series, the first non-trivial term being harmonic.
These quadratic terms often result in Gaussian integrals, which can be
handled analytically. Already the introduction of slightly more complicated term
renders the integrals unsolvable and forces to use stronger approximations.
We are going to confine a polymer in a cylindrical symmetric environment,
hence we introduce a cylindrical symmetric harmonic potential. In the limit of
the weakly-bending rod, which is only justified when the tube is very narrow
(ld ≪ L and/or lp ≫ L, cf. 2.4.1), some calculation are rendered possible since
the inextensibility constraint has not to be taken into account explicitly.