Size of the Blob

2 Size of the Blob Let us first estimate the size of the blob f (Fig. 4.2). Each polymer chain consists of N monomers of size b. The polymer is monodis-perse. For convenience, we use the monomer density p defined in Section 2.4.S.2. It is the number of monomers in a unit volume and related to the mass concentration c by 

At the overlap concentration (defined in Section 1.8), the N monomers in a volume of give the overall monomer density p  

The blob size f is equal to R at p. As the solution becomes more concentrated, the chains become more heavily overlapped with each other. When there are more entanglement points, the blob must decrease its size (Fig. 4.3). 

We make two assumptions to estimate the blob size in the semidilute solution at monomer density p  

The negative exponent on p tells that the blobs become smaller with an increasing p. Note also that f does not depend on N explicitly. It is determined by the monomer density or the mass concentration only, once the polymer is given. 

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In the case of crew-cut micelles, //corona < Rcok and the logarithm in (22) can be expanded up to the term linear in //corona// core, to give / corona/ B — Z/corona/i / = //corona/. The thickneSS Of the corona, //corona, scales as //corona = In the framework of the Alexander-de Gennes blob model [51, 52], the micellar corona (the planar brush) can be envisioned as an array of closely packed blobs with size = 5 /, equal to the average distance between the coronal blocks. We note that a constant size of the blobs implies //corona Na. The number of coronal blobs per chain //corona/ is proportional to the free energy of the interchain repulsion that equals fcorona/kBT = ... 

Employing the blob model for semi-dilute solutions, we define the size of the blob as... 

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 insert between To and Xr. In other words, the 2/3 scaling segment is inserted before the 1/2 segment, as illustrated in Fig. 5.4. 

Note that as concentration increases, the sizes of the blob and of the chain decrease. In the bulk, we recover Floiy s results the interaction completely screened, the size of the blob is the step length, and the chain is ideal. Thus the present model ensures a gradual cross-over from... 

At even higher concentrations, where the volume fractions of polymer and solvent are comparable, the size of the blob reduces to that of the length of a polymer segment Then, the conformation of the polymer chains can be described again by a random walk. Hence, concentrated polymer solutions (and polymer melts) are always at 0 conditions. 

The balancE between three-body repulsion and electrostatic attraction determines the size of the blobs and the number of monomers in them. 

The blob is a conceptual object. Unlike f go, we cannot measure the size of the blob. Later, we will derive the identity, blob size = correlation length. The latter can be conveniently measured in static and dynamic light scattering. 

We can also use these techniques to determine how the radius of an individual coil changes with concentration. Look at Figure 11.3.4. We visualize an individual chain as a succession of blobs , each of molecular weight Mb- Inside each blob the coil is swollen so that the contour length of the coil (and therefore Mb) within each blob is related to the size of the blob by... 

As pointed out above, the statistics inside each blob is the same as in bulk, so that the linear size of the blob Dp scales with the number of monomers g inside... 

The scaling theory discussed above for stars can also be apphed to chains grafted to a hne. Each branch or arm is again treated as a series of spherical blobs. Since the volume accessible at a radial distance r from the grafting line increases, the monomer density decreases and the size of the blobs increases with r. At a distance r from the grafting line, a cylinder of length L has piL blobs of radius (r) covering a cross-sectional area of Lr. Therefore the blob radius,... 

For r>R(., the mesh size of the transient network should be equal to the blob size at distance R,., i.e.. 

In fact, in a recent work on poly(vinylethylene), the size of the Gaussian blob has been found to correspond to about 20 bonds [39]. 

Mayes et ai. (1994) tested this prediction using SANS on PSPMMA and PMMA-dPMMA diblocks in contrast-matched toluene/d-toluene mixtures in the disordered phase. They did not obtain the scaling (eqn 4.14), instead a best fit to the data yielded q = (p°05 (Fig. 4.34). This slower than expected scaling is presently unexplained. However, the expected concentration dependence of the blob size (Mayes et at. 1994)... 

Similar problems are abundant as soon as we leave the region of small momenta and isolated chains. As a final example we consider the semidilute limit. Using the unrenormalized loop expansion in Sect, 5.4.3 we have calculated the first order correction to fip(n). We found a correction of order where c is the segment concentration. The form of this term is due to screening and has nothing to do with the critical behavior treated by renormalization and -expansion. It thus should not be expanded in powers of e. We can trace it back to the occurrence of the size of the concentration blobs as an additional length scale. 

The second term of Eq. (4) (Ornstein-Zernicke function) accounts for the liquid-like contributions from the polymer network and provides the correlation length (mesh size or hydrodynamic blob size) of the PVFA-co-PBVU/silica and PVAm-co-PBVU/silica network meshes [100]. The mesh sizes for PVFA-co-PBVU/silica and PVAm-co-PBVU/silica hydrogel hybrids are listed in Table 2 as a function of temperature. 

These equations can be solved for the size of the tension blobs in terms of the normal size (i o or Rf) and stretched size Rj) of the chains ... 

The size of the real chain confined between plates is again much larger than that of an ideal chain (where R Ri bN ) because the compression blobs of the real chain repel each other. The maximum confinement cor-responds to thickness D of the order of the Kuhn monomer size b. In this case the chain becomes effectively two-dimensional with size... 

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