The Scaling Theory

De Gennes24,54) investigated the adsorption of a flexible polymer on a flat surface from good solvents using the scaling theory. Three different regimes, i.e. dilute, semidilute, and plateau regions, were considered. 

For a single chain of n segments confined in the adsorbed polymer layer with thickness D, the chemical potential fx is given by 

D is independent of n and increases with decreasing 6. Eq. (B-124) is rewritten using Eq. (B-126) as follows  

(B-127) is equated to the chemical potential jub in the bulk solution with concentration Q, we get 

In the plateau region many polymer chains are adsorbed and give rise to an adsorbed layer with thickness D. The chemical potential ju in this region is expressed by 

---

According to the scaling theory of the adsorption transition [2,35], one expects for e near e. in the limit A oo a power law behavior... 

The main predictions of the scaling theory [40], concerning the dynamics behavior of polymer chains in tubes, deal with a number of characteristic times the smallest time rtube measures the interval of essentially Rouse relaxation before the monomers feel the tube constraints significantly, 1 < Wt < Wrtube = and diffusion of an inner monomer is... 

A simulational test [19] of the power law (39) can be seen in Fig. 15(a). As it is clear from the raw data, presented in the insert to Fig. 15(a) the data for zjD 1 settle down to constant values, and a region where zjD is still small and a power law is seen cannot be easily identified. The scaling theory, however, is expected to hold only for distances z that are large on the microscopic scale [74]. Studying such a regime where z is very large in... 

Figure 9. Experimental data for the effective viscosity of the foam bubble regime in Berea sandstone as a function of the foam superficial velocity. The solid line is drawn according to the scaling theory with values of the two sets of parameters e and 6 listed.

Croxton [50] found some differences with respect to the scaling pictures with discontinuities in the bead density at the central core. These results are in agreement with his theoretical calculations with the iterative convolution technique, also for short chains. However, the main features of the scaling theory prediction for the blobs density has been satisfactorily verified by Crest et al. [157] who performed a BD simulation of stars interacting through an LJ potential. The scaling... 

The factor 0.03 is deduced here for s-state and a half-filled band, and simple diagonal disorder (Fig. 1.17). We believe that it has much wider validity, and can be applied, at any rate in a theory of non-interacting electrons (as for instance in a semiconductor) to any form of disorder or for p- or d-states. Our confidence depends on the success of the scaling theory of Abrahams et al. (1979), which will be outlined in Section 13. 

Gai. jMn.jAs. Furthermore, the scaling theory of electronic states near the MIT, discussed in the previous sections, makes it possible to explain the presence of the ferromagnetism on the both sides of the MIT, and a non-critical evolution of 7fc across the critical point (Matsukura et al. 1998b). A comparison between theoretical and experimental data in a wider range of Mn and hole concentrations requires reliable information on the hole density in particular samples, which is not presently available. In appears, however, that in the case of both Gai-jjMnjAs and Im jMnjAs on the insulator side of the MIT, the experimental values of Tc are systematically higher than those expected from the Zener model. 

The experimentally obtained value of the exponent 1.5 is, therefore, in a good agreement with the scaling theory which predicts the exponent of 3/2. 

There is a substantial body of theoretical work on micellization in block copolymers. The simplest approaches are the scaling theories, which account quite successfully for the scaling of block copolymer dimensions with length of the constituent blocks. Rather detailed mean field theories have also been developed, of which the most advanced at present is the self-consistent field theory, in its lattice and continuum guises. These theories are reviewed in depth in Chapter 3. A limited amount of work has been performed on the kinetics of micellization, although this is largely an unexplored field. Micelle formation at the liquid-air interface has been investigated experimentally, and a number of types of surface micelles have been identified. In addition, adsorption of block copolymers at liquid interfaces has attracted considerable attention. This work is also summarized in Chapter 3. 

The scaling theory for spherical polymer brushes due to Daoud and Cotton (1982) (Section 3.4.1) has been applied to analyse the coronal density profile of block copolymer micelles by Forster et al. (1996). If the density profile is of the hyperbolic form r as found by FOrster et al. (1996) for the coronal layer of block copolymer micelles, the brush height scales as... 

Fig. 3.24 Micellar core radius, RB, as a function of Ng for PS-poly(caesium acrylate) (A) and PS-poly(caesium methacrylate) ( ) in toluene (Nguyen el al. 1994). A linear relationship is anticipated by the scaling theories of Zhulina and Birshtein (1985) and Halperin (1987,1990) for type IV micelles.

The theoretical description of excluded volume effects on the adsorption from good solvents is still unsatisfactory. The scaling theory for polymer adsorption has not yet been subject to experimental tests. 

To determine qo we consider the large momentum behavior. In the excluded volume limit the scaling theory predicts (cf. Eq. (9.20))... 

In the present paper, calculations are carried out to explain the restabilization followed sometimes by destabilization, as the electrolyte concentration increases, observed by Stenkamp et al. In these calculations, double-layer, steric, and depletion interactions are taken into account. The steric interaction at various electrolyte concentrations is calculated using the scaling theory. The surface density of the polymer chain was evaluated by using the Sechenov equation for the polymer solubility as a function of electrolyte concentration. 

---