Polymers in Confined Geometry.pdf

3.2. SCALING RELATIONS 25
b
a
d
R ||
Figure 3.3: Minimal extension R || of a chain with stiff segments of size b in a tube of diameter
d.
starting from a totally stretched conformation and not allowing ‘directional flips’
between fore and aft—which are very unlikely for chains with persistence—the
polymer can take a configuration with a minimal end-to-end distance by making
a zig-zag configuration along the tube axis. This is calculated by using the
Pythagoras’ theorem (cf. figure 3.3)
where
Hence:
a = √ b 2 − d 2 = b
min R|| ≈ Na, (3.7)
1 − 1
2
min R|| = √ b 2 − d 2 ≈ L
2 d
+ O
b
1 − 1
2
4
d
. (3.8)
b
2
d
, (3.9)
b
for d ≪ b. This result is true even, when dealing with an ideal phantom chain,
which should not show any influence of a confining environment. This argument
is only due to finite-size effects and particularly interesting in connection with
simulations (cf. section 4.4).
3.2.2 The stiff regime
Now turning to the limit of stiff polymers we can use again a ‘blob’-picture like
argument to arrive at a scaling relation. We know that the end-to-end distance
of a stiff polymer scales as
R
L ∼
⎧
⎨1
for strong confinement d → 0
⎩1
− L
for no confinement d → ∞ and
6lp
L , (3.10)
≪ 1
lp
where the second line is the square root of the series expansion of eq. (2.21).