Excluded volume parameter, definition

Universality and two-parameter scaling in the general case of finite excluded volume, Be comes about by the much more sophisticated mechanism of renormalization. As will be discussed in later chapters (see Chap. 11, in particular) both the discrete chain model and the continuous chain model can be mapped on the same renormalized theory. The renormalized results superficially look similar to expressions like Eq. (7.13), but the definition of the scaling variables iie, z is more com plica led. Indeed, it is in the definition of R ) and z in terms of the parameters of the original unrenormalized theory, that the difference in microstructure of the continuous or discrete chain models is absorbed. 

Let us begin with the estimation of the polymer volume fraction inside the coil formed by one long semiflexible macromolecule. It is well known that this estimation depends essentially on the strength of the excluded volume effect, i.e. on the value of the parameter z = vN /a, where v is the excluded volume of a monomer and a is the spatial distance between two neighbouring monomers. To be definite let us adopt for a moment the model shown in Fig. 1 b. Then, if we choose one segment as an elementary monomer, v 6. (see Eq. (2.4)) and a i.e. z p Consequently, the excluded volume effect is pronounced at N p and negligible at N [Pg.77]

Although, the true density of solid phase p=m/Vp (e.g., g/cm3) is defined by an atomic-molecular structure (/ ), it has become fundamental to the definition of many texture parameters. In the case of porous solids, the volume of solid phase Vp is equal to the volume of all nonporous components (particles, fibers, etc.) of a PS. That is, Vp excludes all pores that may be present in the particles and the interparticular space. The PS shown in Figure 9.17a is formed from nonporous particles that form porous aggregates, which, in turn, form a macroscopic granule of a catalyst. In this case, the volume Vp is equal to the total volume of all nonporous primary particles, and the free volume between and inside the aggregates (secondary particles) is not included. 

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