Scaling law theories

The scaling law theory is for the most part beyond the scope of this work, but several results will be used in this chapter. In the scaling law theory, the transition from one regime (i.e., dilute to semidilute... 

As discussed by de Gennes," scaling law focuses attention on the exponents and refrains from the determination of any prefactory constants, which are usually much more difficult to determine. Some predictions of scaling law theory for polymers of high molecular weight in both good solvents and 9 solvents in the semidilnte regime are summarized in Table 7.3. 

Scaling law theories are concerned with the universal properties of polymer molecules. These originate in their global characteristics. The detailed local properties are ignored. This concept is well illustrated by the radius of gyration, Rg, of a polymer chain. Flory has shown that / , is given by... 

Scaling law theories are especially concerned with transitions from one regime to another. Suppose there exists a transition at some critical value x of the parameter x where the variable 5 changes its form. Then the fundamental postulate of scaling law theory is that S can be written ... 

Obviously, So is the form adopted by S in regime (i). Such a transition is represented schematically in Fig. 4.10. Note that whilst scaling law theory can... 

In the semi-dilute range, is seen to be a function of the polymer concentration and the chain length. The scaling law theories assume that in the semi-dilute range This challenging assumption, which appears... 

The consonance noted above between the predictions of scaling law theories and the results of experiments suggests that the assumptions underlying the theories are correct. Mattice (1981), however, has criticized the basic assumptions of the blob concept because they are not verified by rotational isomeric state calculations that incorporate the excluded volume term, admittedly in a somewhat empirical fashion. A major difficulty envisaged by Mattice in the simple application of the blob concept lies in its failure to account for the consequences of the repulsive interaction of the subchain with atoms elsewhere in the main chain. 

Scaling law theory applied to attached polymer chains 4.6.4.1. Segment density profiles... 

Lax et a/. (1981) have applied scaling law theory to a free polymer behaving as a self-avoiding walk in a thin slab of thickness h. This limits the span of the polymer chains leading to distortions in the direction parallel to the slab plane. The rms end-to-end distance of a self avoiding walk of isolated chains is given by... 

Further experimental proof of the occurrence of interpenetration resides in the verification of the scaling laws for concentrated polymers, reviewed in Section 4.6.3.5. Scaling law theories assume that polymer chains interpenetrate in the semi-dilute regime and their predictions appear to be confirmed experimentally. 

Dependence upon the molecular weight of free polymer It is possible to provide a crude estimate of the exponent involved in the dependence of Ae critical volume of free polymer (vj ) required for flocculation on the molecular weight of the free polymer. This is in the spirit of scaling law theory. 

It should be stressed, however, that according to scaling law theory, neither Cz nor Cz represent well-defined concentrations but rather concentration regions that may be rather broad. 

Scaling law theories were developed by de Gennes (Al) and others to provide novel solutions to a series of physical problems, including some in polymer science. Scaling law is concerned with exponents, sometimes called universal properties, rather than coefficients, sometimes called local properties (A2). Scaling law starts with known relationships, with special concern with transitions between regimes of physical behavior. 

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