Polymer Physics

The fraction of charged monomerspon the polyelectrolyte backbone is highly
sensitive to the environmental factors such as temperature, strength of the electric
field, ion concentration and pH value (Barrat and Joanny 1996 ). In the polyelectro-
lyte solutions, the Coulomb force on the polymer chain plays an important role in
determining the chain conformation. The Coulomb energy between two charge
units is

Eco

e^2

4 pe 0 el

(4.49)

wheree 0 is the dielectric constant of vacuum,eis the relative dielectric constant of
solvent,lis the distance between two charge units. At a certain distance, the
Coulomb energy between two charge units becomes comparable with the thermal
energykT. This certain distance is calledBjerrum screening length, as given by

lB¼

e^2

4 pe 0 ekT

(4.50)

This length scale characterizes the distance between two charges of the same
species allowed by the thermal fluctuations. In other words, when the charge
densitypon the polymer chain becomes too high, for instance,p>e/lB, the
counter-ions will accumulate around the chain to maintain the effective charge
density at constant e/lB. Such a phenomenon is calledManning condensation of
counter-ions(Manning 1969 ).
The fraction of charged monomers on the chain determines the Coulomb energy
of single chain, as given by

EcokT

lBðnpÞ^2

R

(4.51)

Here,nis the chain length, andRis the average coil size. The electrostatic
repulsion between the same species of ions on the repeating units leads to coil
expansion. This expansion energy is balanced with the entropy loss associated with
chain conformationEel~R^2 /(nb^2 ), then the total free energy change of the coil is

F¼EcoþEel¼kT½

lBðnpÞ^2

R

þ

R^2

nb^2

Š (4.52)

Fig. 4.10 Illustration of a macro-ion and its surrounding counter-ions in polyelectrolyte solutions

60 4 Scaling Analysis of Real-Chain Conformations