Characteristic ratios

In this section, we study the effects of local interaction on chain properties. A real chain has a fixed bond length (/ = 0.154 nm) and a fixed bond angle (0 = 109.47°) between subsequent carbon atoms. The internal rotation experiences a potential energy which depends upon the rotational angle. It is generally described by (1.9). 

The internal hindered rotation affects the chain statistics in many ways, but the fundamental nature of a Gaussian chain, such that its mean square end-to-end distance and radius of gyration are proportional to the molecular weight of the chain, remains unaltered, although the rotional potential energy modifies the proportionality constants. Therefore, to study the proportionality constant, we introduce the characteristic ratio 

Let us first consider the free rotation model. The free rotation model has a mean square end-to-end distance 

For a large n, the second term can be neglected. The characteristic ratio is then given by the Eyring formula [12]  

When the potential is not imiform, the characteristic ratio takes the more general form 

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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... 

SO tliat tire characteristic ratio can be evaluated from tire plateau value of i (EXCLUDED-VOLUME EFFECTS... 

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... 

At PicArsn (Ref 19), the fast neutron activation approach for detection of expls in suitcases was extended to the activation of both nitrogen and oxygen using two 7-ray detector stations in sequence. After 14 MeV neutron irradiation, the baggage is first monitored for 6.1 MeV 7-rays from the l60(n,p),6N reaction (7.5 sec half-life), followed by measurement of the 10 min 13N. Because expls are also rich in oxygen and have characteristic ratios of N/O, it was felt that this approach would increase the probability of detection with a corresponding decrease in the false alarm rate... 

The stabilizing influence of small amounts of B (M/B > 0.25) in the voids of the metal host lattice varies with the mode of filling (partial or complete) of the interstitial, mostly O, sites and whether the compounds develop from the binary-intermetallic host lattice. The structures of B-rich compounds (M/B < 4) are mainly determined by the formation of regular, covalent B polyhedra (O, icosahedron) and the connections between them (B frame structures). Typical metal (M) borides therefore are found within a characteristic ratio of metal to boron 0.125 < M/B < 4. 

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... 

Fig. 5.11. Characteristic ratio of PE plotted vs. temperature, for N = 20 effective monomers. Both versions of the mapping, using the exact four-bond procedure and the approximate two-bond procedure, are included. From [32]...

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... 

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

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... 

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. 

Lipids are generally identified in paint samples by evaluating characteristic ratio values of FA amounts and comparing them with naturally or artificially aged reference paint layers. Molar or weight contents are obtained after quantification based on calibration curves. 

Such a model ignores restrictions on the bond angles and dihedral angles imposed by the local chemical structure. This may be taken care of empirically by defining a characteristic ratio, Cm, where... 

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