Flory theorem

Polymeric chains in the concentrated solutions and melts at molar-volumetric concentration c of the chains more than critical one c = (NaR/) ] are intertwined. As a result, from the author s point of view [3] the chains are squeezed decreasing their conformational volume. Accordingly to the Flory theorem [4] polymeric chains in the melts behave as the single ones with the size R = aN112, which is the root-main quadratic radius in the random walks (RW) Gaussian statistics. 

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. 

Due to such densely packed molecularly interpenetrated structures, rubbers are incompressible under deformation. Each chain takes a Gaussian conformation following the Flory theorem for screened excluded-volume interaction. On the basis of these characteristics, we can derive the elastic properties of rubbers from a microscopic point of view. 

For three dimensional situations, neutron experiments probing a few labeled (deuterated) chains in melt of identical (hydrogenated) chains, have shown quite convincingly that the chains are ideal and gaussian as expected fiom the Flory theorem. - The radius is R = a, and the local concentration due to one labeled chain is of order N// N a . This local concentration is small this implies that there are many chains overlapping to build up the total concentration (a ) in the melt, and that Fig. II.6a applies. 

Experiments of this type have been carried out on partially deuterated polystyrenes (using quenched phases from the melt) by J. P. G>tton and co-workers. These experiments give us precise information oil the local correlations between chains in a polymer melt. Also, because of the simplicity introduced by the Flory theorem, this is one of the few cases where the scattering diagrams can be computed accurately. The method is described in Chapter X. Here we present the results only, in qualitative terms. 

The main asstrmptions used concern the Gaussian character of the chains and the absence of restrictions imposed by otber chains to tbe conformation of a given chain. They are based on the Flory theorem, which states that the statistical properties of polymer chains in a dense system are equivalent to those for single ideal chains. The reason is that in a imifoim, amorphous substance all the conformations of a certain chain are equally likely in a sense that they couespond to the same energy of interaction with other chains, because the surrotmdings of each emit are roughly the same. 

The particularly simple relationships between the average end-to-end distance of the random coil and the chain length that are derived in section 2.4 are valid under the ideal solution conditions referred to as theta conditions. The dimension of the unperturbed polymer chain is only determined by the short-range effects and the chain behaves as a phantom chain that can intersect or cross itself freely. It is important to note that these conditions are also met in the pure polymer melt, as was first suggested by Flory (the so-called Flory theorem) and as was later experimentally confirmed by small-angle neutron scattering. 

This equation can be applied to any real polymer under theta conditions considering that any such chain can be represented by a hypothetical equivalent chain with n freely jointed links, each link being of length / (section 2.4.4). Equation (2.93) is, according to the Flory theorem, also applicable... 

Small-angle neutron scattering (SANS) of labelled (deuterated) amorphous samples and rubber samples detects the size of the coiled molecules and the response of individual molecules to macroscopic deformation and swelling. It has been shown that uncrosslinked bulk amorphous polymers consist of molecules with dimensions similar to those of theta solvents in accordance with the Flory theorem (Chapter 2). Fernandez et al (1984) showed that chemical crosslinking does not appreciably change the dimensions of the molecules. Data on various deformed network polymers indicate that the individual chain segments deform much less than the affine network model predicts and that most of the data are in accordance with the phantom network model. However, defmite SANS data that will tell which of the affine network model and the phantom network model is correct are still not available. 

The approximate treatment of the bonding constraints in Eq. 3 may be motivated by recourse to the Flory theorem [3,4], which states that in polymer melts it is impossible to discern whether a pair of nonbonded nearest-neighbor united atom groups belongs to different polymer chains or to distant portions of the same polymer molecule. Thus, in the lattice model description of polymer systems, the excluded volume prohibition of multiple occupancy of a site is more important than the consequences of long-range chain connectivity. Based on the Flory theorem, we introduce the zeroth-order mean-field average... 

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