Gaussian-coil networks

A restriction will be made to polymer chains without strong and specific intersegmental forces, such as may exist in proteins and many other macromolecules. The reason is that the elasticity of networks composed of such chains cannot even be approached from the Gaussian coil point of view, since the chains are helical or at least partly so. At the moment no good theoretical treatment of these "proteinlike systems is available. 

The thermodynamic affinity of cyclohexane to polystyrene is known to increase with temperature and, naturally, increasing the temperature must further raise the volume of the polystyrene networks in cyclohexane. There is, however, an additional point we should consider. The plot of Q vs. temperature exhibits a steplike discontinuity at around 30°C (Fig. 1.14). This discontinuity, resemhling very much a -transition, is located 3-5°C below the -temperature for linear polystyrene in cyclohexane and about 8°C above the -point for star-shaped polystyrene macromolecules. This phenomenon is outside the scope of the questions discussed here, but, naturally, the first assumption of the authors [143] seems to be very logical, according to which the discontinuity reflects a transition from Gaussian coil to a supercoiled compact structure on cooling the swollen gel below that temperature zone. 

In an equilibrium undeformed network, subchains prefer to reside in the state of Gaussian coils, which can be realized by a maximum number of conformations, thus providing the highest entropy to the system. When the stress extending the network is applied, the subchains are elongated. The potential extensibility of the coil can be estimated as N/VN, because the length of a fully extended chain is proportional to N, while the most probable end-to-end distance in the coil is proportional to Therefore, the subchain consisting of 100 units... 

We will derive an expression for the stress in a rubber network as a function of extension ratio. The starting point is the probability distribution function for a random Gaussian coil ... 

The simplest model of rubber-like behaviour is the phantom network model. The term phantom is used to emphasize that the configurations available to each strand are assumed to depend on the positions of the junctions only. Consequently, the configurations of one chain are independent of the configurations of neighbouring strands. For many purposes, the strands can be treated as Gaussian random coils. Even in this simplest case, an exact solution is not a trivial task as will be outlined in Sect. 3. 

And at last, the third and the most fundamental factor is the ehange of nanocomposite structure at the introduction of particulate filler in high-elastieity polymeric matrix. As Balankin showed [9], classical theory of entropic high-elastieity has a number of principal deficiencies due to non-fulfilment for real rubbers of two main postulates of this theory, namely, essentially non-Gaussian statisties of real polymeric networks and lack of coordination of postulates about Gaussian statistics and incompressibility of elastic materials. Last postulate means, that Poisson s ratio v of these materials must be equal to 0.5. As it is known [10], Gaussian statistics of macromolecular coil is correct only in case of its dimension Dj=2.0, i.e., for coil in 0-solvent. Since between value Df and fractal dimension... 

One of the first successful theories of polymer physics was developed for rubber elasticity, and we now briefly outline the essential ideas. It is first assumed that the deformation occurs without changing the sample volume. It is also assumed that the chain segments between crosslinks adopt the Gaussian conformation of an unperturbed coil. The deformation is taken to be affine, i.e. it is the same at the molecular level as at the macroscopic scale. If the sample is deformed by extension ratios Xi, X2 and X3 in three different directions, its dimensions change by these fractional amounts. In an affine deformation the coordinates of the end point of a network chain move by the same factors, i.e. from (x, y, z) to (Xix, kiy, Xsz). 

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