Unperturbed Gaussian conformations

The conformation of polymer chains in an ultra-thin film has been an attractive subject in the field of polymer physics. The chain conformation has been extensively discussed theoretically and experimentally [6-11] however, the experimental technique to study an ultra-thin film is limited because it is difficult to obtain a signal from a specimen due to the low sample volume. The conformation of polymer chains in an ultra-thin film has been examined by small angle neutron scattering (SANS), and contradictory results have been reported. With decreasing film thickness, the radius of gyration, Rg, parallel to the film plane increases when the thickness is less than the unperturbed chain dimension in the bulk state [12-14]. On the other hand, Jones et al. reported that a polystyrene chain in an ultra-thin film takes a Gaussian conformation with a similar in-plane Rg to that in the bulk state [15, 16]. 

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). 

The RPA is a mean field approximation that neglects contributions from thermal composition fluctuations and that assumes the chain conformations to be unperturbed Gaussian chains. The last assumption becomes visible from the Debye form factor in the first two terms, which for Vp, = are in accordance with Eq. 7, while the third term involves the FH interaction parameter. 

The idea that a chain in a melt is effectively a free chain without selfinteraction was first clearly expressed by Flory and is often called the Flory theorem. At first sight it is paradoxical how can the presence of other chains allow a chosen chain to take up conformations that it could not take up if the other chains were absent The answer is that it cannot. The unperturbed chain is only equivalent to the real chain in terms of its Gaussian statistics. This is nevertheless very important, because it is the statistical properties of the chains and their link with the entropy that largely determine some of the properties of the corresponding materials. 

The Gaussian chain provides an exceQent description of conformations in unperturbed polymer liquids and in the glasses obtained cooling them. 

In either the melt state or in a 0-solvent solution, linear flexible polymers adopt Gaussian statistics, and their average conformation is described as a random walk. Consequently, the ratio of their unperturbed mean-square... 

Region Ifo appears for semifiexibie chains (p > 1) only at p > 1, the overlap concentration c for Gaussian chains is small as a result, the individual chain conformation remains unperturbed by interactions with other drains in spite of significant overlap in the region IIq. 

The Gaussian chain provides an excellent description of conformations in unperturbed polymer liquids and in the glasses obtained by cooling them. In real polymer products, however, the molecules often do not have the random shape of a Gaussian chain. Forces exerted on the liquid during forming cause... 

Allegra has shown that the probability distribution for more general linear combinations of the skeletal bond vectors of an unperturbed chain is also Gaussianly distributed. In particular, if we decompose the conformation into the Fourier modes, 7 (p), defined by... 

For all the afore-mentioned computations of Kx values Semiyen et al. used Eq. (5.24). The unperturbed mean square end-to-end distance of the linear precursors were calculated via the matrix algebraic methods of Flory and Jemigan [84, 85] using rotational isomeric state models based on detailed structural information [86]. However, Semiyen et al. [62, 63] also improved and applied another mathematical approach to the calculation of Kx, the so-called Direct Computational Method . The JS theory is limited to polymers obeying Gaussian statistics and cycles, free of enthalpic interactions. The Direct Computational Method does not need such restrictions [87-90]. The distances between terminal atoms of chains are calculated for all discrete conformations defined by the rotational isomeric state model. Any correlation between the directions of terminal bonds involved in the cyclization process can be investigated and their effect on Kx assessed. It can take into account favorable and unfavorable correlations between the directions of terminal bonds, as well as any excluded volume effect. Semiyen demonstrated [62, 63, 72] that the Direct Computational Method yields more realistic Kx values for small cyclic oligomers. 

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