Polymer Physics

Sincexis a local property of the chain, irrelevant to chain lengthn, thus
1+2a¼0, and we obtaina¼1/2. Substituting again (4.68) and (4.69) into
(4.70), we have

xLð

C^

C

Þ^1 =^2 ðpcCÞ^1 =^2 (4.72)

Polyelectrolyte chain of a length scale larger thanxwill appear as unperturbed
chain due to the screening of the electrostatic repulsion,

Rxð

n

g

Þ^1 =^2 (4.73)

Here each blob contains the number of monomers

gCx^3 (4.74)

Substituting (4.72) and (4.74) into (4.73), one obtains

Rn^1 =^2 ð

pc

C

Þ^1 =^4 (4.75)

With the increase of concentrations, the electrostatic repulsion will be gradually
screened out, and the correlation lengthxwill decay down to the segment lengthl
connecting two consecutive beads. Whenx¼l, the electrostatic repulsion between
two beads is completely screened, and the volumex^3 can accommodate only one
bead. According to (4.65) and (4.67), we havel~pc^1. From (4.72), we reach the
critical concentration

Cbpc^1 l^2 pc (4.76)

WhenC>Cb, the electrostatic repulsion between two beads is completely
screened, and the chain segment connecting two beads will disappear. According
to (4.74), we have

mgCx^3 (4.77)

From the charge balance on the bead surface, we have already obtained (4.65),
i.e.m~pc^2. From (4.77), we further obtain

xðpc^2 CÞ^1 =^3 (4.78)

In the meantime, the polyelectrolyte chain formed purely by a string of blobs
without any in-between connecting chain segment will exhibit the unperturbed
chain conformation. Substituting (4.65) and (4.78) into (4.79), one derives

4.3 Single-Chain Conformation in Polyelectrolyte Solutions 65