Chains characteristic ratios

Coo being the chain characteristic ratio. Consequently, the statistical weight of a closed loop comprising n chain bonds (i.e., r = 0) is given by... 

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 ... 

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>. 

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... 

Qx>B characteristic ratio of a chain with all side groups replaced by hydrogens... 

For poly(methylene), an exclusion distance (hard sphere diameter) of 2.00 A was used to prevent overlap of methylene residues. The calculation reproduced the accepted theoretical and experimental characteristic ratios (mean square unperturbed end-to-end distance relative to that for a freely jointed gaussian chain with the same number of segments) of 5.9. This wps for zero angular bias and a trans/gauche energy separation of 2.09 kJ mol". ... 

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...

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. 

Recently a very detailed study on the single chain dynamic structure factor of short chain PIB (M =3870) melts was undertaken with the aim to identify the leading effects limiting the applicability of the Rouse model toward short length scales [217]. This study was later followed by experiments on PDMS (M =6460), a polymer that has very low rotational barriers [219]. Finally, in order to access directly the intrachain relaxation mechanism experiments comparing PDMS and PIB in solution were also carried out [186]. The structural parameters for both chains were virtually identical, Rg=19.2 (21.3 A). Also their characteristic ratios C =6.73 (6.19) are very similar, i.e. the polymers have nearly equal contour length L and identical persistence lengths, thus their conformation are the same. The rotational barriers on the other hand are 3-3.5 kcal/mol for PIB and about 0.1 kcal/mol for PDMS. We first describe in some detail the study on the PIB melt compared with the PDMS melt and then discuss the results. 

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... 

Table I. Measured and Computed Room Temperature Characteristic Ratio and Temperature Coefficients for Cellulosic and Amylosic Chains...

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 ... 

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