Polymer Physics

The repulsive energy between the blobs should be comparable with the energy
level ofkTfor thermal fluctuations. Accordingly, the osmotic pressure of the semi-
dilute solution at the theta point becomes

P

kT

x^3

kTC^9 =^4 (4.47)

In practice, from the dilute solutions to semi-dilute solutions, the scaling law of
the osmotic pressure changes rather gradually, as roughly given by

P

kTC



1

MN

þC^5 =^4 (4.48)

This relationship has been well verified by experiments (Rubinstein and
Colby 2003 ).
When a long chain is mixed with middle-length polymer chains, one can imagine
that within a certain length scale, the volume exclusion of chain units in the long chain
will be screened out by the middle-sized “solvent”. Above this length scale, the long
chain exhibits the chracteristics of expanded coils. Theoretical estimation shows that
this length scale corresponds to the square of middle chain lengths. In other words, if
the length of the long chain cannot go beyond the square of the middle chain lengths,
its volume exclusion will be completely screened out by the solvent chains, and the
long chain will exhibit the ideal-chain characteristics (De Gennes 1979 ).
In thermal concentrated solutions, polymer chains in a good solvent behave
similarly to those in an athermal solvent, while in a poor solvent, polymer chains
experience a phase separation with the coexistence of polymer-rich and polymer-
poor phases. We will give more descriptions about the phase separation behaviors
in Chap. 9.

4.3 Single-Chain Conformation in Polyelectrolyte Solutions

In a polar solvent, a polyelectrolyte chain can dissociate the counter-ions on the side
groups, and becomes amacro-ionor acharged polymer, as demonstrated in
Fig.4.10. The macro-ion chain is surrounded by small counter-ions. If the charge
valence of the counter-ions is larger than one, they can effectively cross-link the
macro-ions to form polyelectrolyte gels.
Bio-macromolecules such as DNA, RNA and proteins exhibit the properties of
polyelectrolytes in aqueous solutions. The migration of fragments hydrolyzed from
bio-macromolecules in the aqueous gel can be oriented by the weak electric field.
Long chains drift slowly, short chains drift fast, and their difference in the speed
results in a characteristic spectrum. This is the principle ofgel electrophoresis.Asa
fundamental method in gene engineering, gel electrophoresis has been widely applied
in the identification and analysis of DNA and protein characteristic sequences.

4.3 Single-Chain Conformation in Polyelectrolyte Solutions 59