Kuhn effective segment

The parameter Cr expresses the effectiveness of the hindrance release by segment fluctuation and varies between 0 and 1. An empirical expression for it as a function of the Kuhn statistical segment number N is given in Sect. 8. Although Cr contributes only to the correction terms in de/Le, fr(de/Le) changes from 1 to 2.56 in the range of allowable values of de/Le. Thus, the factor fr (de/Le) is more important than the factor f (de/Le) in Dx. 

It does not affect the exponents in the equation (i.e.. the dependence of on N), however, but simply introduces a prefactor. This suggests a different approach, where we consider the number of adjacent bonds whose combinations of allowed rotations essentially behave like a freely jointed unit when taken collectively. We would then have Nx effective segments each of length lp known as the Kuhn segment length (Figure 8-36), defined in Equation 8-12 ... 

There is quite a large range of Kuhn lengths for real polymers. Rather modest values of about 1 nm are typical for simple synthetic chains, whereas DNA s effective segment stretches 100 nm. (This is a huge number in molecular world, considering that an atom s size is of the order of 0.1 nm )... 

Thus, the effective interaction has the form of dficay (serwning), and f is the screening length in the units of Kuhn s segment (cf. Equation 5.1 256)... 

Taking into account that the effects of short-range ordering, e.g., of fixed valence angles and hindrance of internal rotation [3, 4], do not change the shape of the distribution function and the correlation Rg-N but increase Ro, that can be interpreted in terms of increase of a statistical length of an equivalent Kuhn chain segment [5]. 

In good solvents, the mean force is of the repulsive type when the two polymer segments come to a close distance and the excluded volume is positive this tends to swell the polymer coil which deviates from the ideal chain behavior described previously by Eq. (1). Once the excluded volume effect is introduced into the model of a real polymer chain, an exact calculation becomes impossible and various schemes of simplification have been proposed. The excluded volume effect, first discussed by Kuhn [25], was calculated by Flory [24] and further refined by many different authors over the years [27]. The rigorous treatment, however, was only recently achieved, with the application of renormalization group theory. The renormalization group techniques have been developed to solve many-body problems in physics and chemistry. De Gennes was the first to point out that the same approach could be used to calculate the MW dependence of global properties... 

Values of B calculated from the ordinate intercepts are shown in Fig. 23 as a plot of B/(2q)3 against the number of the Kuhn segments N. For N<4, the data points for the indicated systems almost fall on the solid curve which is calculated by Eq. (78) along with Eqs. (43), (51), (52), and Cr = 0. A few points around N 1 slightly deviate downward from the curve. Marked deviations of data points from the dotted lines for the thin rod limit, obtained from Eq. (78) with Le = L and de = 0, are due to chain flexibility the effect is appreciable even at N as small as 0.5. The good lit of the solid curve to the data points (at N 4) proves that the effect of chain flexibility on r 0 has been properly taken into account by the fuzzy cylinder model. 

These theoretical data show that the influence of the octatomic siloxane ring is observed at n=1 only. Saturation is observed already at transition from n=5 to n=10, and increase of the quantity of =SiO- groups causes no effect on the coil size. For the fraction of copolymer 3 of the structure I and n = 5, experimental values of the Kuhn segment and /nM0 are shown in Table 4. 

Fig. 5.8. Top Transverse relaxation curve calculated for a given cross-link density (diamonds) and multiparameter fits The Gauss-Lorentz fit (broken line) agrees well at short times. The biexponential fit (continuous line) agrees well at long times. Bottom T2 relaxation curves of the CH group for nine differently cross-linked samples of unfilled SBR samples. The effective number N,. of Kuhn segments per cross-link chain varies between 9.43 and 15.56 corresponding to vulcameter moments between 16.5 and 1 dNm. By renormalization of the time axis the master curve (top) has been obtained.

Figure 11.15. Effects of input material parameters on stress-strain curves of elastomers under uniaxial tension, as calculated by the theory of rubber elasticity with finite chain extensibility. G denotes the shear modulus, while n denotes the average number of statistical chain segments (Kuhn segments) between elastically active network junctions, (a) Engineering stress a as a function of draw ratio X, as calculated by using Equation 11.41. (b) True stress (simply equal to aX for an elastomer) as a function of true strain [In (A,)].

Figure 11.18. Effects of crosslinking on the brittle fracture stress of styrene-divinylbenzene copolymers. Note the catastrophic embrittlement at very high crosslink densities (i.e., at an average number of repeat units between crosslinks less than the Kuhn segment length of 8 repeat units). The data point (not shown) for the uncrosslinked limit (polystyrene) is at (°°,41).

The notions of functional self-similarity and scaling arise very naturally in polymer science and have found applications in this field for many years. Thus, a linear polymer chain with excluded-volume interactions is a perfect example of a physical object to which scaling should be applicable, as a subdivision of the entire chain into a collection of blobs or Kuhn macrosegments [9,14]. One of these segments may be envisioned as a fragment of sufficient length to ensure that its statistical properties are effectively independent from the remainder of the chain. 

The effective coupling function for the Kuhn segment is obtained by solving the equation... 

We recently performed calculations of the partition function for a randomly jointed chain with hard-sphere excluded-volume interactions [21], namely, the same model for which the swelling factor was calculated in Section IV.A. The effective coupling function for the two-bond K = 2) Kuhn segment is displayed in Figure 5.8. It is apparent that the spectrum of fixed points becomes quasicontinuous for g > 2. (An examination of the roots of the equation 3Q K" g) — [3Q(K = 0 confirms this... 

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