Characteristic ratios definition

The formal definition of the characteristic ratio Cro, which is a more popular descriptor of polymer chain conformation than o, is given by Equation 12.2, where "lim" denotes "limit" n is the number of bonds along the shortest path across the chain backbone and nj is the number of times the i th type of bond, which has a bond length of lj, occurs along this shortest path. 

The result in equation (4.8) for a freely jointed chain leads to the definition of the characteristic ratio C of a real polymer molecule composed of n bonds. This is... 

The characteristic ratio of the poly(methylene) chain, which might be considered to be the simplest polymer molecule, as a function of chain length is shown in Fig. 4.4 for the simple models considered thus far. All save the simple freely jointed chain display end effects, manifest by an increase in Ci with chain length at small n, which will not be considered further. In what follows, only the asymptotic limit of the characteristic ratio for 00 (Coo) will be discussed. The values of for the freely jointed chain, the freely rotating chain and the chain with independent bond rotational potentials increase in value from 1 through 2 to ca 3-5. The latter value is, however, only ca one-half of the experimentally determined value of Coo=6-9 for poly(methylene). This serious discrepancy points to the fact that the bond rotational potentials are definitely not independent, i.e. the conformation of bond i depends upon the conformations of bonds (/—I) and (/-i-1). 

Polymer mechanical properties are one from the most important ones, since even for polymers of different special-purpose function a definite level of these properties always requires [20]. Besides, in Ref [48] it has been shown, that in epoxy polymers curing process formation of chemical network with its nodes different density results to final polymer molecular characteristics change, namely, characteristic ratio C, which is a polymer chain statistical flexibility indicator [23]. If such effect actually exists, then it should be reflected in the value of cross-linked epoxy polymers deformation-strength characteristics. Therefore, the authors of Ref [49] offered limiting properties (properties at fracture) prediction techniques, based on a methods of fractal analysis and cluster model of polymers amorphous state structure in reference to series of sulfur-containing epoxy polymers [50]. 

Strictly speaking, the F value is not an indicator of the local ordering degree for polymer structure, since clusters are formed by segments of different macromolecules, but it can be an indicator of mutual penetration of macromolecular coils. As has been shown in papers [22, 100], the same role can be played by the characteristic ratio C. If this assumption is correct, a definite correlation between F and must be observed. The data of Figure 1.30 show that such a correlation is really observed for nine amorphous and semi-crystalline polymers (the F value is calculated for T= 293 K) [99]. 

Since an atom of a given element gives rise to a definite, characteristic line spectrum, it follows that there are different excitation states associated with different elements. The consequent emission spectra involve not only transitions from excited states to the ground state, e.g. E3 to E0, E2 to E0 (indicated by the full lines in Fig. 21.2), but also transisions such as E3 to E2, E3 to 1( etc. (indicated by the broken lines). Thus it follows that the emission spectrum of a given element may be quite complex. In theory it is also possible for absorption of radiation by already excited states to occur, e.g. E, to 2, E2 to E3, etc., but in practice the ratio of excited to ground state atoms is extremely small,... 

In recent years, a dependable dipole function for He-Ar, last column of Table 4.3, has been obtained [278] which we compare with the universal dipole function mentioned [23], Fig. 4.5. The He-Ar interaction potential is one of the better known functions [13] and suggests Rmj = 6.518 bohr. Both functions were normalized to unity at the separation R = 5 bohr in the figure. The comparison shows that at small separations the logarithmic slope of the most dependable dipole function is roughly one half that of the universal p, and p diverges rapidly from p(R) for R — o. Similar discrepancies have been noted for other rare gas systems (Ne-Ar, Ne-Kr, and Ar-Kr [152]). Even if for these other systems the dipole function is not as well known as it is for He-Ar, it seems safe to say that for the rare gas mixtures mentioned the induced dipole function is definitely not identical with the universal function at the distances characteristic of the spectroscopic interactions the universal dipole function is not consistent with some well established facts and data. We note that the ratio of // (/ ) and the He-Ar potential is indeed reasonably constant over the range of separations considered (not shown in the figure). 

Phase diagrams from freezing point depressions show true compound formations for simpler amides—e.g., water-N-methylacetamide forms a compound at a mole ratio of 2 to 1, water-N,N-dimethylacetamide at 3 to 2 and 3 to 1, and water-N-methylpyrrolidone at 2 to 1. The heats of mixing and heat capacities at 25°C. of a number of water-amide systems were determined. All mixing curves were exothermic and possess maxima at definite mole ratios, while the heat capacities for the most part show distinct curvature changes at the characteristic mole ratios. Both experimental results point to the stability of the particular complexes even at room temperature. This is further supported by absolute viscosity studies over the whole concentration range where large maxima occur at these same mole ratios for disubsti-tuted amides and N-substituted pyrrolidones. 

Heat Capacities. The heat capacities (Cp in cal./mole-°C., not shown here) exhibit some unusual features, in that they show marked curvature changes at the characteristic mole ratios found thus far for each of the respective amides or pyrrolidones, again rendering it likely that definite stoichiometric complexes occur in the amide-water mixtures. 

It can be shown that the ratio v3lvg is equal to the ratio of polymer packing densities coefficients in the amorphous and crystalline states, KJKC at Tg, because, by definition, Ka = NA V /va and Kc - NA Vi/yC)where vj is the Van der Waals volume of the chain repeat unit. The calculated values of (ATc)g correlate with the characteristic chain parameter a/o, the relationship between them being expressed by a linear equation... 

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