Polymer Physics

wherebis the size of chain units. From the minimum of this free energy with
respect toR, i.e. let its first derivative toRequal to zero, we can obtain the scaling
law of polyelectrolyte coil sizes as (De Gennes et al. 1976 )

Rnp^2 =^3 ðlBb^2 Þ^1 =^3 (4.53)

In much earlier time, Kuhn et al. have already obtained a more accurate
expression with a logarithmic correction (Kuhn et al. 1948 ), as

Rnp^2 =^3 ðlBb^2 lnnÞ^1 =^3 (4.54)

Assuming that the electrostatic interaction potential around the chain follows the
Poison equation approximately,

@uðrÞ

@r

¼ 4 plBcðrÞ (4.55)

wherec(r)is the counter-ion concentration at a radial distancerfrom the chain, and
further assuming that the counter-ions locally follow the Boltzmann distribution,

cðrÞ¼c 0 expðuðrÞÞ (4.56)

one can obtain the famousPoisson-Boltzmann equation, as given by

@uðrÞ

@r

¼lDH^2 expðuðrÞÞ (4.57)

Here definesDebye-Hu ̈ckel screening length

lDH^2 ¼ 4 plBc 0 (4.58)

Equation (4.57) can be solved under specific geometric conditions (Fuoss
et al. 1951 ).
When salts are added into the polyelectrolyte solutions, the effective interaction
potential between the ions becomes

UðRÞ¼kT

lB

R

eR=lDH (4.59)

Here the Debye-Hu ̈ckel screening length is

lDH¼ð 4 plBIÞ^1 =^2 (4.60)

4.3 Single-Chain Conformation in Polyelectrolyte Solutions 61