Polymer Physics

Rxð

n

m

Þ^1 =^2 n^1 =^2 ð

pc

C

Þ^1 =^3 (4.79)

Since the chain conformation appears as unperturbed in the concentrated
solutions, the coil sizes will not depend on the concentration any more. With further
increase of concentrations,C!1,x~m1/3, and eventually the electrostatic repul-
sion between charged monomers will be completely screened. In the end, the
polyelectrolyte chain will behave like a charge-neutral polymer chain in highly
concentrated solutions. Correspondingly, the coil sizes will increase in a sudden,
leading to an increase of characteristic relaxation time as well as the intrinsic
viscosity, and appearing as a “gelation” process, as demonstrated in Fig.4.13
(Dobrynin and Rubinstein 2005 ).

4.4 Single-Chain Conformation Under External Forces..........

4.4.1 Stretching

If a stretching force is imposed on the two chain ends of a polymer in a good
solvent, the single chain will response with a deformation along the direction of the
force. On the other hand, thermal fluctuations tend to restore the local chain
conformation to the coil state without any stretching, as illustrated in Fig.4.14.
Let’s assume the existence ofN/gblobs, each with a linear size satisfyingr¼bg3/5.
The overall end-to-end distanceRfhas been homogeneously distributed in each
blob with a weak disturbance,

Rf¼r

n

g



R^5 =^3

r^2 =^3

(4.80)

where the sizeR¼N3/5b corresponds to the original coil size without any
stretching. Then, we obtain

Fig. 4.13 Illustration of the
concentration-dependent
scaling laws for coil sizes of
the polyelectrolyte chain
shown as a bead string in a
poor solvent

66 4 Scaling Analysis of Real-Chain Conformations