Polymer Physics

SubstitutingR~m1/3into the equation above, the critical charge density on the
chain to stabilize each bead is thus,

pcm^1 =^2 (4.65)

On the other hand, the surface free energy of the chain connecting two neigh-
boring beadsglb^2 (the chain lengthlwith monomer sizeb) is also balanced by the
charge repulsive energy between two beads (epcm)^2 /(el). Similarly, by minimizing
their total energy with respect tol, we obtain

lpcm (4.66)

Inserting (4.65) into (4.66), we further obtain the chain length connecting two
consecutive beads along the chain

lm^1 =^2 (4.67)

Here the total amount of monomers on the chain isn. If we neglect the
monomers on the connecting chains, the total number of beads is roughlyn/m.If
we further neglect the contributions of bead size, the contour length of the chain
becomes

L

n

m

lpcn (4.68)

The hydrophobic polyelectrolyte chain appears rigid due to the electrostatic
repulsion along the chain. In the concentrated solutions, the electrostatic repulsion
will be gradually screened due to interpenetration of polyelectrolyte chains
(Dobrynin and Rubinstein 2001 ). According to (4.68), the critical overlap concen-
tration for the transition from the dilute solution to the concentrated solution is

C^ 

n

L^3

pc^3 n^2 (4.69)

WhenC>C, similar to the treatment of neutral polymers in semi-dilute
solutions, we can use the blob model to describe the polyelectrolyte segment
holding the electrostatic repulsion in semi-dilute solutions. Assuming the blob
size as a characteristic lengthxto maintain the chain rigidity, with reference to
the critical overlap concentrationC,

x

L

ð

C

C^

Þa (4.70)

Substituting (4.68) and (4.69) into the equation above, we obtain

xn^1 þ^2 a (4.71)

64 4 Scaling Analysis of Real-Chain Conformations