Scaling Law Theories and Applications

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. 

The theory supposes that there exists a transition at some critical value x of a parameter x such that the variable S changes its form. The fundamental postulate of scaling law theory can be written, 

At the critical crossover volume fraction from dilute to semidilute, (p v, the entire volume is just taken up by the molecules. Therefore 

A more comprehensive treatment (A3) yields cpov where the 

As noted above, in a dilute good solvent Rgj where Rg is independent of polymer concentration within the dilute concentration range. Using the subscript s for semidilute and d for dilute, the scaling law can be written 

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