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Gaussian chain anisotropy

For a network of Gaussian chains having the same number n of links, uniaxially stretched by an amount L/Lo = A., the assumptions of affine displacement of jimction points and initial Gaussiein distribution of end-to-end vectors allows one to calculate the optical anisotropy of the network by integrating Eq.lO over the distribution of end-to-end vectors in the stretched state. By taking Treloar s expansion [11] for the inverse Langevin function, the orientation distribution function for the network can be put into the form of a power series of the number of Unks per chain ... [Pg.261]

When the worm-like chain is long, it becomes a Gaussian chain and, according to Eq. (35), its anisotropy is given by... [Pg.120]

Figure 15 shows the dependence of the chain anisotropy (with respect to the anisotropy of the Kuhn segment fiA) on its relative length x = L/a according to Eqs. (39) and (40). At x—> 0 the curves have the slope 0.5 and rapidly attain an a mptotic limit corresponding to the anisotropy of the Gaussian chain. [Pg.121]

Curve I describes the dependence of y on x obtained for kinetically rigid worm-like chains disregarding the dependence of p on x. The asymptotic limit of Curve I is 0.833. .. (instead of y = 1 for Curve 4) because in Reference the anisotropy of the Gaussian chains is taken to be (3A/2 rather than 3/3A/5 used for plotting Curves 1 -4. [Pg.127]

The difference between nematic and isotropic elastomers is simply the molecular shape anisotropy induced by the LC order, as discussed in Sect. 2. The simplest approach to nematic rubber elasticity is an extension of classical molecular mbber elasticity using the so-called neo-classical Gaussian chain model [64] see also Warner and Terentjev [4] for a detailed presentation. Imagine an elastomer formed in the isotropic phase and characterized by a scalar step length Iq. After cooling down to a monodomain nematic state, the chains obtain an anisotropic shape described by the step lengths tensor Ig. For this case the stress-strain relation can be written as ... [Pg.199]

I.3.2.2 Anisotropy The Gaussian chain is isotropic when averaged over many conformations and orientations. In the crudest approximation, we can regard it as a sphere of radius R. The instantaneous shape of the chain, however, does not look like a sphere. We will examine its anisotropic shape here. [Pg.27]

The large difference in the shear mechanical anisotropy and in the chain anisotropy between the two types of NE suggests that the nature of their elasticity is not the same. Since Gaussian elasticity is characterized by a decrease of G when the... [Pg.55]

In the limiting case of a Gaussian coil (x— ) the anisotropy of a worm-like chain in a system of coordinates of its middle element is given by... [Pg.121]

Equations (20) and (23) show that the average rotational friction coefficient W is proportional to the radius of gyration both for extended and symmetrical conformations. Hence, it is proportional to h (compare Eqs. (3) and (3 )p. 98). According to Eq. (46), optical anisotropy in the random chain conformation is also proportional to h in the Gaussian range (h/L -4 1). Hence, a Gaussian coil obeys the following equation... [Pg.126]

It is important to emphasize that the SOR is not the inevitable consequence of fundamental physical principles rather, it is a very plausible hypothesis, which has extensive experimental support for polymer solutions and melts. In other words, there is no reason to assume that the SOR is valid under all possible flow conditions or for all possible polymer liquids. Some situations under which the SOR is expected to fail are mentioned in the next section. Many constitutive relations for solutions and melts predict that the SOR will hold, but even this apparent generality is somewhat misleading. The derivation of an SOR starts at a measurable molecular property, the optical polarizability of an isolated molecule a, and leads to a macroscopic refractive index tensor n, in a nontrivial way several substantial assumptions are necessary. Most rheological models (for flexible chains) that proceed to an SOR assume the derivation of Kuhn and Gritn (1942) for the polarizability anisotropy of a Gaussian subchain and thus in a sense make the same assumptions for the optical half of the SOR (Larson, 1988). Therefore differences between constitutive relations and their predictions for an SOR usually stem from differences in the calculation of t. [Pg.395]

Segmental anisotropy (a — 02) of a chain molecule may be determined experimentally from measurements of flow birefringence in the solution of a polymer with a sufficiently high molecular weight, so that the conformation of its molecules would correspond to that of a Gaussian coil. [Pg.2221]


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See also in sourсe #XX -- [ Pg.26 ]




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