<½rðtpÞrð 0 Þ^2 >ZR^2 pZt
2 = 3
pZ (5.38)Thus we can obtain that, in the time window oft 0 <t<tZ, corresponding to
the size scalen6/5b^2 within the single coil,
<½rðtÞrð 0 Þ^2 >Zt^2 =^3 (5.39)The exponent here (2/3) is slightly larger than that of the Rouse chain (1/2),
implying again a faster diffusion. This scaling law can well describe the diffusion of
a single polymer in dilute solutions.
In concentrated solutions, with the increase of the polymer concentration, the
screen effect of hydrodynamic interactions is enhanced due to the interpenetration
of polymer chains. We can assume that thehydrodynamic screening lengthxhis
close to the screening length of volume exclusion of monomersx, as given by
xhx (5.40)Employing the blob model for semi-dilute solutions, we define the size of the
blob as
xg^3 =^5 C^3 =^4 (5.41)The monomers inside each blob move in a range smaller thanxh, where the
conditions for the non-draining mode are maintained. Therefore, the Zimm model is
applicable.
txx^3 C^9 =^4 (5.42)In contrast, beyond each blob, the motion range of monomers is larger thanxh,
where the conditions for the free-draining mode are restored due to the interpene-
tration of polymer chains. Therefore, the Rouse model is applicable.
tRtxðn
gÞ^2 n^2 C^1 =^4 (5.43)The size of the blob ranges from the size of monomers to the whole chain,
depending upon polymer concentrations. Therefore, the dynamic scaling law for the
single short chain in the semi-dilute solutions is to inserttxbetweent 0 andtR.
In other words, the 2/3 scaling segment is inserted before the 1/2 segment, as
illustrated in Fig.5.4.
84 5 Scaling Analysis of Polymer Dynamics