Polymer Physics

Due to the screening effect in the volume exclusion of polymer chains, single-
chain conformation in the concentrated solutions will exhibit the size scaling
similar to the ideal-chain model, as

Rn^1 =^2 (4.22)

However, we wish to note that near the critical overlap concentration, the coil
size is actually dependent on the polymer concentration in the concentrated poly-
mer solutions.
We know that when the polymer volume fractionC<C*, the coil size in dilute
solutions exhibits

Rn^3 =^5 (4.23)

At the critical overlap concentration, assuming that chain units are homo-
geneously distributed within each coil,

C^ ¼Cint/

n

R^3

n^1 ^3 n (4.24)

For the expanded single coil,n¼3/5, thus one can obtain

C^ n^4 =^5 (4.25)

WhenC>C*, the coils start to interpenetrate into each other. Daoud et al.
proposed a blob model to describe such states of polymer chains (Daoud et al.
1975 ). As demonstrated in Fig.4.8, they treated the single chain as a string of liquid
droplets referred as blobs, which is similar to a pearl-necklace. The blob size is

Fig. 4.7 Illustration of the spatial distribution of single-chain units in the concentrated solutions
(solid curve). Below the screening length, the pair distribution behaves like a single chain in dilute
solutions (long dash curve); and beyond this characteristic length, it behaves like an ideal single
chain (short dash curve)

4.2 Single-Chain Conformation in Polymer Solutions 53