The characteristic ratio

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 a real chain can be considered crudely in terms of a time averaged spherical model (polymers adopt at any instant a shape that is more sausage like than spherical but undergo rotational random motion). Accordingly, C is the ratio of the average cross-section of the real chain compared with that of the corresponding random flight chain. Some values of Coo for common polymers are shown in Table 4.1. 

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It is not possible to apply (C2.1.1) down to the level of monomers and replace by the degree of polymerization N and f by the sum of the squares of the bond lengths in the monomer because the chemical constitution imposes some stiffness to the chain on the length scale of a few monomer units. This effect is accounted for by introducing the characteristic ratio defined as C- — The characteristic ratio can be detennined... 

The characteristic ratio approximately indicates how extended the chain is. For the free 3 rotating chain the characteristic ratio is given by ... 

The dilute solution properties of copolymers are similar to those of the homopolymer. The intrinsic viscosity—molecular weight relationship for a VDC—AN copolymer (9 wt % AN) is [77] = 1.06 x 10 (83). The characteristic ratio is 8.8 for this copolymer. 

Thus, if the ratio AT//3 is constant, then the behavior shown in Fig. 12 could be described by the net solution. For many polymers, the characteristic ratio is around 7-10, the ratio Mo/j is the molecular weight per backbone bond (ca. 30-50) and will not vary extensively, b = 1.54 A and the density is about 1 g/cm such that the parameter is nearly constant. Since K is not very sensitive to the polymer properties, Eq. 6.5 is considered to describe the observed fracture behavior shown in Fig. 12. However, the data are not expected to fall on the straight line due to differences in K and j6 for each polymer. As a specific test case, consider... 

One decisive test is how faithfully the geometrical characteristics of single chains in the melt can be reproduced. Figure 5.11 shows that the characteristic ratio CN = (R2)/Np c has a temperature dependence that is very similar to... 

An experimental test of the scaling model requires a selective variation of the two scaling variables of the model, i.e. the lateral chain distance and the chain stiffness. The Kuhn length /K depends on temperature via the characteristic ratio Cw the lateral chain distance s can be varied via the volume fraction 4>. 

Since both the temperature dependence of the characteristic ratio and that of the density are known, the prediction of the scaling model for the temperature dependence of the tube diameter can be calculated using Eq. (53) the exponent a = 2.2 is known from the measurement of the -dependence. The solid line in Fig. 30 represents this prediction. The predicted temperature coefficient 0.67 + 0.1 x 10-3 K-1 differs from the measured value of 1.2 + 0.1 x 10-3 K-1. The discrepancy between the two values appears to be beyond the error bounds. Apparently, the scaling model, which covers only geometrical relations, is not in a position to simultaneously describe the dependences of the entanglement distance on the volume fraction or the flexibility. This may suggest that collective dynamic processes could also be responsible for the formation of the localization tube in addition to the purely geometric interactions. 

The molecule is either fully flexible or semi-flexible. The fully flexible chains are generally harder to crystallize than semi-flexible chains [35]. In the latter part of the paper (Sect. 5), where we discuss crystallization from the melt, we consider a semi-flexible chain, the flexibility of which is adjusted to reproduce the characteristic ratio of real polyethylene. We there make the... 

Figure 5 is an ORTEP computer plot of the first 50 backbone carbons in a typical chain. Only the fluorine atoms of the sidechains are shown. The various hard sphere exclusions conspire dramatically to keep the fluorines well separated and the chain highly extended even without introducing any external perturbations. The characteristic ratio from the computer calculations is about 11.6 from data for poly(p-chlorostyrene), CR = I l.l, poly(p-bromostyrene), CR = 12.3, and poly(styrene), CR = 10.3 (all in toluene at 30°C), we expect the experimental value for the fluoro-polymer to be in the range of 10 to 12. 

Table 2.1 illustrates the magnitudes of the characteristic ratio found for typical polymers in dilute solution. The relatively simple polyethylene oxide), (PEO), chain is fairly flexible whereas the cellulosic chain has... 

Table 2.1 Values of the characteristic ratio for various chains...

For the segment length we take where the characteristic ratio... 

In a real chain segment-segment correlations extend beyond nearest neighbour distances. The standard model to treat the local statistics of a chain, which includes the local stiffness, would be the rotational isomeric state (RIS) [211] formalism. For a mode description as required for an evaluation of the chain motion it is more appropriate to consider the so-called all-rotational state (ARS) model [212], which describes the chain statistics in terms of orthogonal Rouse modes. It can be shown that both approaches are formally equivalent and only differ in the choice of the orthonormal basis for the representation of statistical weights. In the ARS approach the characteristic ratio of the RIS-model becomes mode dependent. 

As explained above, towards shorter scales a more realistic description of the chain dynamics must include the stiffness of the chain. The influence of the stiffness that can be expected from the characteristic ratio of PIB was calculated according to both the ARS model and bending force models. For the mode... 

Figure 4. The characteristic ratio Coo calculated as a function of temperature T for values of the glycosidic valence angle 116 p < 124 . The dashed curve describes experimental results for cellulose triacetate.22...

A further expansion of the average dimensions of the coil results when one assumes that rotation around single bonds is not free but is still independent of the rotation around the adjacent bonds. Let us take as an example a polyethylene chain On the base of each of the two cones of formula 70, three positions are identified, T, G", and G, differently populated according to the energy difference E = Eq — Ej and the temperature. The characteristic ratio is then written ... 

It is seen that the values of the characteristic ratio estimated by the Stodc-mayer-Fixman method scatter about the Brant-Flory value, tending to increase systematically with monomer molar weight This trend is not in accord with the theoretical prediction (96, 97) that the characteristic ratios of polypeptides having side-chains longer than that of polyalanine should become constant. We... 

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