Polymers in Confined Geometry.pdf

24 CHAPTER 3. POLYMERS IN CONFINED GEOMETRY
d
blobs
Figure 3.2: ‘Blob’ picture scaling: Inside the blobs of size d the polymer behaves as if it were
free. The self-avoidance is taken into account by treating the blobs as hard-walled, stacking
them linearly.
The derivation of the intermediate scaling regime can be done most illustrative
by the “blob picture” introduced by de Gennes. For a tube of diameter d the
polymer can expand up to this diameter as if it were free, using g < N monomers.
Using the real-chain end-to-end scaling, eq. (2.34), g can be calculated (cf. figure
3.2)
d ∼ (b 2 v) 1/5 g 3/5 ⇒ g ∼ d 5/3 (b 2 v) −1/3 . (3.4)
The key assumption of the de Gennes scaling argument is, that, due to selfavoidance,
the ‘blobs’ behave as hard spheres. Hence N/g-‘blobs’ must be stacked
linearly along the tube:
R|| ∼ N
d ∼ Nv1/3
g
2/3 b
. (3.5)
d
Using the three different monomer volumes vi introduced in section 2.5 we obtain
R||1 ∼ L
b 2/3
, (3.6a)
d
R||2 ∼ L bh
d2 1/3 , (3.6b)
R||3 ∼ L
h 2/3
. (3.6c)
d
Again the relation R||1 is mostly found in the literature, but the disk-like distribution
of monomers R||2 is actually the right one to use for polymers with thickness
h = 0.
Actually R|| should better be called apparent parallel end-to-end distance Ra||,
as explained in section 4.6, because as the figure 3.2 suggests, the scaling is valid
for the maximal parallel extension 3 .
Approximation for strong confinement
When looking at very strong confinement, a point is reached where the tube
diameter d is smaller than the segment length b = L/N. In this case, when
3 Since we are not going to include self-avoidance effects, for the reasons discussed on page
18 and in section 4.4.2, we do not expect to see the scaling derived here. When stretching for
flexible chains is seen, then due to finite-size effects.