Blob picture

We do not intend to give an overview over all results of scaling theory here. Rather we concentrate on topics relevant for the bulk behavior of normal polymer solutions. We discuss in particular the concentration dependence, introducing the blob -picture (Sect. 9.1). Temperature dependence is discussed in Sect. 9.2. The results are summarized in the Daoud-Jannink diagram [DJ76] which separates parameter space into several regions, where different characteristic behavior is expected. 

Again we can rewrite this in terms of quantities introduced in the blob picture ... 

Ebert U, Baumgartner A, Schafer L (1996) Universal short-time motion of a polymer in a random environment Analytical calculations, a blob picture, and Monte Carlo results. 

In the above node-link-blob picture, the percolation cluster is self-similar up to a length scale in the sense that starting from the length scale the links contain blobs (and the dangling ends) which, in turn, are composed of links and blobs (and the dangling ends) up to the lowest scale (of the lattice). This self-similarity extends up to infinite scale at the percolation threshold (where becomes infinitely large). 

Failure current for p near Pc We saw in the preceding section that the failure current decreases when p approaches pc and becomes zero at the threshold. We shall propose a different determination of the failure current from the node-link-blob picture of the infinite cluster (see Section 1.2.1(d)). Instead of using the current I (current fiowing through the sample), we shall consider the tension V between the two electrodes. The relation between I and V is given by V = RI where the resistance of the sample is dependent on p,... 

Consider the Pincus blob picture of stretching an ideal chain, discussed in... 

V = jl under good and theta-solvent conditions, respectively). Hence, the blob picture enables one to derive the power law for the radial decay in polymer density ... 

The blob picture of a semidilute solution of polymer stars is schematically presented in Fig. lb. The peripheral (contracted) regions of the star coronae are envisioned as a sea of blobs of constant size, which corresponds to a constant polymer concentration in this region. In contrast to this, within radius p(c) < R, the structure of the corona of individual stars is preserved (a system of growing... 

The structure and the basic thermodynamics of micelles formed by amphiphilic block copolymers with a PE coronal block A can be analyzed using the blob model. However, the ionization of block A in a polymeric amphiphile introduces long-ranged repulsive interactions in the corona of a micelle. As a result, the blob picture for the micellar corona has to be modified, as explained in this section. 

Figure 3. A blob picture of the polymer under a force. Though drawn as sphere, the z-dircction is elongated with isotropy in the trans ersc direction.

The above analysis, done routinely for polymers, relies on the fact that there is only one length scale in the problem, namely, the size of the polymer. If we are entitled to do the same for the disorder problem, namely only one scale, i o IV", matters, then the blob picture goes through in toto. The chain breaks up into blobs and the blobs align as dictated by the force. Each blob is independent and the polymer inside a blob is exploring its environment like a directed polymer pinned at one end. The probabihty distribution is therefore given by Eq. (64) which for d = 1 is... 

Figure 5.8. The blob picture of a chain in a semi-dilute solution. The average blob diameter is given by the correlation length some of the interacting chains are shown as dashed lines.

We can now use the blob picture of a single chain to get the scaling relation for the radius of gyration. On average each blob contains g chain units of length a and because excluded volume conditions prevail within the blob, then... 

Figure 6.3. The blob picture of a polymer brush in the semi-dilute regime.

Another derivation of eq. (I.S3) is based on a blob picture. The chain behaves as a sequence of blobs of diameter D. Inside each blob the effects of the boundaries are weak. The number gp of monomers per blob is still given by the three-dimensional law go = D/a. Successive blobs act as hard spheres and pack into a regular one-dimensional array. Thus f ii = N/go D in agreement with eq. (1.53). 

In the simple blob picture, the free ends are assumed to be within the last blob. Analytic solutions of the SCF equations, both for polymers in solu-... 

Figure 2. Blob picture for a random chain adsorbed from a dilute solution. Near the wall the chains are stretched. On the outer side the chains build a self-similar layer. 

To proceed it is necessary to obtain an explicit expression for Fchain- o simple approaches available are the Flory approximation and the blob picture. Within the Flory approximation the polymers are viewed as ideal, uncorrelated chains. Accordingly, the number of monomer-monomer interactions per site scales as Since the volume per chain is aL, Fchain/ v/a ) aL where vkT is the second virial coefficient. The elastic penalty of a Gaussian chain is F /kT iP /R where Ro is the unperturbed radius of the coil. Altogether... 

Within the blob picture the brush is viewed as a slab of semi-dilute solution. It is thus envisioned as an array of close packed blobs of size consisting of... 

This picture, of a chain confined to a capillary, is of great utility. As we shall discuss, it guides the generalization of the blob picture to non-planar geometries and the analysis of dynamical properties. Before we address this issues, it is useful to consider the structure of a brush immersed in a poor solvent. 

This result may be interpreted as the osmotic work done by a brush of imit width as it overspills, ttA, where ir/kT (jy is the osmotic pressure in a brush consisting of ideal chains. Since the blob picture leads to tt kTja l with no modification in L, this suggests that the blob or scaling form of r is specified by two length scales, A and f and is given by ra/kT ... 

Finally, recent SANS experiments by Auroy and Auvray probed the behavior of brushes immersed in a solution of homopolymers in a good solvent. Their results confirm the theoretical expectations. In particular, they observed the shrinking of the brush due to increased screening. Also observed was the associated decrease in the correlation length or blob size. Both results are of special interest as direct confirmation of the blob picture. 

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