Polymers in Confined Geometry.pdf

16 CHAPTER 2. POLYMER MODELS
frequently called phantom chains. But in reality polymers are charged objects in
a solution surrounded by charges. Therefore the interaction between the polymer
segments is very complex and depends on the solution. But often one is entitled
to replace these interaction by a simplified hard core interaction. The next step in
the direction of a more realistic model is then to incorporate self-avoidance effects
as a steep short-ranged repulsive potential between the segments (of the order of
the microscopic polymer thickness). The polymer is not allowed to intersect with
itself 7 . For detailed description of models with ‘realistic’ features confer e.g. [39].
The influence of the self-avoidance/excluded volume is visible in particular
for flexible/long (i.e. a large ratio of L/lp) chains since they fold back and forth
freely, therefore crossing themselves very often. As a result the polymer will tend
to be more extended than a phantom chain due to the reduced number of allowed
configurations (entropic penalty) in the strongly collapsed state. In contrast for
stiff polymers the effect is mostly negligible since it is unlikely for them to fold
back onto themselves.
At variance with the description of the ideal phantom chain as a random walk,
the self-avoiding random walk can be used to investigate the properties of these
chains. But we want to focus on the derivation of the basic scaling relation for
the end-to-end distance.
The considerations of the “entropic penalty” can be used to derive a scaling
relation similar to eq. (2.6) for the ideal chain. This argument dates back to P.
Flory and is described in detail in [8] and [39]. It considers a correction to the
free energy of the ideal chain, eq. (2.8), that is due to the excluded volume which,
in turn, depends on the number of monomer-monomer contacts. Let v be the
mean volume occupied by a monomer depending on the segment length b and
the thickness of the chain d. The monomer density φ of a chain of linear size R
which occupies a sphere of about the volume R 3 is
φ ∼ Nv
. (2.32)
R3 The entropy loss for an exclusion is of order kBT for each degree of freedom
(equipartition theorem). Since there are N monomers the free energy correction
is
F ∼ kBT v
N 2
. (2.33)
R3 Additionally using the free energy of the ideal chain, eq. (2.8), one can minimize
the sum of these contributions with respect to the end-to-end distance R. We
obtain the scaling result
R ∼ (b 2 v) 1/5 N 3/5 . (2.34)
7 There exists a so called θ-temperature (depending on the solution) where the segmentsegment
interaction is exactly balanced by effects from the surrounding solution. Then the
description of an ideal chain is valid.