The Flory Theorem

A fundamental assumption that underlies the polymer solution theory in this chapter is the factorization of the chain partition function in Equations (31.4) and (31.7) into a product of two terms. One term, which depends on factors 

High polymer concentrations are predicted to decrease the populations of all the conformations uniformly. Is this remarkable theorem correct To a very good approximation, neutron-scattering experiments first performed in the 1970s show ed that it is [1]. A labeled chain in a bulk polymer medium has the same average radius as that expected for an ideal chain, which we describe in Chapter 32. 

A polymier molecule is much larger than a typical small solvent molecule, often by more than a thousandfold. This difference in size has important consequences for the thermodynamics of pol mier solutions. First, the coUigative attraction of a polymer solution for soWent small-molecule molecules is much higher than would be expected on the basis of the mole fractions. Rather the attraction is proportional to the volume fraction. Second, polymers are typically not miscible with other pol miers because of small mixing entropies. Third, the size as Tnmetry between a polymer and a small-molecule solvent translates into an asymmetry in miscibility phase diagrams. 

Concentrations mole fractions x versus volume fractions p. Consider a solution of polymers P and small molecules s, with = l p = 0.5. The pol mier chain length is N = 1000. 

Flory-Huggins mixing free energy. A polymer A with chain length Na = 1000 is mixed with a small molecule B. The volume fractions are (f)A = 0.3 and (pp = 0.7, and Xab =0.1. The temperature is 300 K. Using 

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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. 

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... 

Batch Polymerization. Batch polymerization with this mechanism was first treated by Flory (19) using a statistical development. The same results were obtained by Biesenberger (8) using a kinetic analysis with an analytical solution. This was also one of the cases treated by Kilkson (35) using Z-transforms. In the simple cases, his result reduces to the Flory, or random, MWD with the dispersion index of 2. In more complex cases, he solves directly for the moments of the distribution. The Z-transform is probably the most powerful tool for solving condensation MWD problems the convolution theorem allows the nonlinear product terms in the kinetic equation to be handled conveniently. 

Our discussion here explores active connections between the potential distribution theorem (PDT) and the theory of polymer solutions. In Chapter 4 we have already derived the Flory-Huggins model in broad form, and discussed its basis in a van der Waals model of solution thermodynamics. That derivation highlighted the origins of composition, temperature, and pressure effects on the Flory-Huggins interaction parameter. We recall that this theory is based upon a van der Waals treatment of solutions with the additional assumptions of zero volume of mixing and more technical approximations such as Eq. (4.45), p. 81. Considering a system of a polymer (p) of polymerization index M dissolved in a solvent (s), the Rory-Huggins model is... 

In the Flory model, no ring formation is allowed, and g>-— 0, so that from Eq. (104) it follows that yc —thus the critical dilution disappears. In real gelations, such disappearance of yc never occurs because of the presence of the finite cyclization probability. The critical dilution is a general theorem in real gelations. 

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. 

Relationship 93 expresses Flory s c theorem of gels (Flory, 1953 de Gennes, 1979). This is the very re.eison why, in gel, the end-to-end distance of a network chain is established equal to the screening length in sernidilute solutions of linear macromolcculos (cf. section 3.3). 

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