Polymer Physics

defined as the screening lengthx. Each single chain contains a number of blobsg.
Within the length scale ofx(r<x), the volume exclusion between chain units
dominates the interactions, and then

xg^3 =^5 (4.26)

Beyond the screening lengthx, the volume exclusions have been completely
screened out due to the interpenetration of polymer coils. Therefore, the chain can
be regarded as an unperturbed coil formed by blobs with the unit sizex, and then

Rxð

n

g

Þ^1 =^2 (4.27)

wherexis actually the correlation length of volume exclusions.
By substituting (4.26) into (4.27), one obtains

Rxn^1 =^2 x^5 =^6 ¼n^1 =^2 x^1 =^6 (4.28)

We know that the higher concentrationCcauses more interpenetration, leading
to the decrease ofx. With the reference of the critical overlap concentrationC*,we
assume

x

R^

ð

C

C^

Þa (4.29)

wherea<0. SinceR/n3/5, andC/n4/5, one obtains

xCan^3 =^5 þ^4 =^5 a (4.30)

xis a local property, irrelevant to the chain lengthn, thus

3

5

þ

4

5

a¼ 0 (4.31)

Fig. 4.8 Illustration of the
blob model of a single
polymer chain in the
concentrated solutions. The
screening length defines the
size of each blob

54 4 Scaling Analysis of Real-Chain Conformations