Law of Self-dilution

The implications of this Law of Self-Dilution with regard to reaction rates and cycUzation reactions are discussed in Chaps. 5 and 7. 

In this context the experiments of Hocker et al. [96], Rempp et al. [97], Roovers et al. [98]. and other research groups [99] should be recalled. Those authors obtained high yields of cyclic polymers from preformed difunctional polymers (e.g., bisanionic polystyrene) by polycondensation with difunctional reaction partners at an IMC of 10 mol/L. The final concentration of chains in experiments (A) and (B) are by a factor 10 lower, and chain growth without any cyclization is absolutely unlikely In summary, the existence of the critical IMC is unproven, it is in contradiction to the results of Kricheldorf et al. [46, 47], it is in conflict with the calculations of Gordon et al. or Stepto et al. (see Chap. 7), and it is in conflict with the law of self-dilution combined with the RZDP ... 

Finally, it should be emphasized that all the theoretical considerations presented above also apply to syntheses of hyper branched polymers as discussed in Chap. 11. Cyclization limits the chain growth according to Eqs. (7.13) or (7.14), it reduces the dispersity compared to Flory s calculations (see Sect. 4.3) and the law of self-dilution (Eq. (7.11)) is also valid together with the RZDP. Furthermore, the existence of a permanent competition of cyclization and chain growth in KC polycondensations is decisive for a proper understanding of non-stoichiometric poly condensations as discussed in Chap. 8. 

The interdiffusion coefficient D is mea,sured when polymer is redistributed under a gradient of its concentration. Theoretical consideration of this process is ba.sed on the laws of unequilibriiim thermodynamics (sec subsections 2.4.3 and 3.3.1). If a labelled macro-molecule is present among their ensemble (see subsection 3.6.2) its motion is characterized by the self-diffusion coefficient (D2). Roth those quantities coincide in dilute solutions D D i Do, but the coefficient D rises with polymer concentration while D decreases significantly. 

Quantitation of equilibrium constants of self-association needs the fraction of free monomers that are present in dilute solutions of known concentration. Suitable assumptions were made such as the intensity of the free hydroxyl band as a measure of the free monomers. The intensity from absorbance of the isolated hydroxyl band, I is related to the absorptivity coefficient, e, the concentration, c, and the path length I by the Beer-Lambert law ... 

Although these examples demonstrate the feasibility of using calculated values as estimates, several constraints and assumptions must be kept in mind. First, the diffusant molecules are assumed to be in the dilute range where Henry s law applies. Thus, the diffusant molecules are presumed to be in the unassociated form. Furthermore, it is assumed that other materials, such as surfactants, are not present. Self-association or interaction with other molecules will tend to lower the diffusion coefficient. There may be differences in the diffusion coefficient for molecules in the neutral or charged state, which these equations do not account for. Finally, these equations only relate diffusion to the bulk viscosity. Therefore, they do not apply to polymer solutions where microenvironmental viscosity plays a role in diffusion. 

Fick s first law, in the form given by Equation 5.12, allows us to define the tracer- and the self-diffusion coefficients. Diffusion of a tracer isotope is the case when a diffusing atom, which is marked by their radioactivity of by their isotopic mass (see Figure 5.1) [7], is introduced in an extremely dilute concentration in an otherwise homogeneous crystal with no driving force [4], In this case, the tracer gradient of concentration will give rise to a net flow of tracer atoms. Consequently,... 

For practical purposes, the difference between GC and DH theory is that the former treats flat surfaces (characteristic for big colloidal particles), as compared to ions in DH theory, and that GC does not impose any restriction on the potential, whereas the DH approximation invokes low potentials. Unlike the GC-case, the DH theory is self-consistent in that the potential of mean force and the mean potential are identical. So the DH equations have the status of exact limiting laws for low potentials and infinite dilution. 

The concentration in the adsorbed layer decreases away from the surface as (f) (z/b) for exponent 0.588. This power law concentration profile in an adsorbed polymer layer was proposed by de Gennes and is called the de Gennes self-similar carpet. This profile of adsorbed polymer can be described by a set of layers ot correlation blobs with their size of order of their distance to the surface z (see Fig. 5.11). The self-similar concentration profile starts at the adsorption blob ads in the first layer. The self-similar profile ends either at the correlation length of the surrounding solution if it is semidilute or at the chain size Rf bN if the surrounding solution is dilute. 

All the theoretical work described so far has assumed conformational ideality. That is, the intramolecular pair correlations are presumed to be independent of fluid density (and composition in an alloy) and can be computed based on a chain model that only accounts for short-range interactions between monomers close in chemical sequence. This assumption can fail spectacularly in dilute good solution where the effective intrachain monomer-monomer interaction is repulsive in a second virial coefficient sense. " For such good solvent conditions, the polymer mass/ size relationship no longer obeys the ideal random walk scaling law R but follows the self-avoiding walk (SAW) law i /V" with v = ... 

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