Florys theory

Flory s theory concerns the distribution of excluded volume u between 2 polymer molecules over a volume V of solution. If there is an independent volume exclusion on the part of the individual molecules, the partition function of the system is 

Expanding the logarithmic terms in series and manipulating for the purpose of 

Note that n2/V = cNa/M, where c is the concentration in grams per unit volume, and N is Avogadro s number. Equation (9.20) can be rewritten as 

While Eq. (9.22) clearly relates A2 to u and M, it vaguely indicates that the value of A2 for a polymer system in solution decreases with the increasing value of M. It is vague because the theory has not specified yet the significance of u. To interpret u, the model needs to be improved. 

As a first approximation, it is convenient to neglect self-avoidance and use a Gaussian approximation to estimate the fractal dimension df which relates the number of monomers N of the fractal to Rq, N. Within this 

Here the subscript o indicates that the excluded volume interactions are absent. For a linear polymer 1, eq. (9,43) reproduces the result for a Gaussian chain, namely dfo = /vo = 2. Levinson found that it also works well for triangular Sierpihski gaskets (J= 2In 3/In 5 1.365) embedded in dimension 3 d 8. For percolation clusters at pc, d 4/3, ° and dfa = 4, which has been confirmed by Grest and Murat ° using MD. For tethered membranes, d = D = 2 and dfo = oo. A more detailed analysis by Kantor et al. found that Rgo In iV which they confirmed by MC simulations. 

Self-avoidance can then be introduced at the level of Flory theory as for linear polymers by balancing the elastic (entropic) free energy of the phantom object without self-avoidance with the mean-field estimate of the excluded volume interaction,  

Thus even in the case when self-avoidance is important, df depends only on the spectral dimension d and the dimension of space d. 

Despite the fact that the Flory argument is a simple mean-field theory, it works very well for a number of cases, often producing estimates for df which differ from the exact result by only a few percent or less for d d c- Here d c = 4d/ 2 - d) is the upper critical dimension above which self-avoidance is irrelevant. For a hnear polymer eq. (9.44) works extremely well for d duc = 4. In three dimensions, the theory predicts u = fdf= i/5, which is very close to the best renormaUzation groups estimates of 0.588. Percolation clusters at the percolation threshold are another example where eq. (9.45) works well. Since d 4/3 in all dimensions,eq. (9.45) predicts that 2( jf-h 2)/5, independent of the 

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The reversible aggregation of monomers into linear polymers exhibits critical phenomena which can be described by the 0 hmit of the -vector model of magnetism [13,14]. Unlike mean field models, the -vector model allows for fluctuations of the order parameter, the dimension n of which depends on the nature of the polymer system. (For linear chains 0, whereas for ring polymers = 1.) In order to study equilibrium polymers in solutions, one should model the system using the dilute 0 magnet model [14] however, a theoretical solution presently exists only within the mean field approximation (MFA), where it corresponds to the Flory theory of polymer solutions [16]. 

Various modifications of the Flory theory [4] are usually applied to describe the uncharged gels. Their crosslinking density can be simply calculated from the swelling degree using Eqs. (3.1) and (3.2) or analytical expressions for the Mc value (see, for example, Ref. [124]). 

For the first non electrostatic term, such a dependence can be calculated from the classical Flory theory and the value of the theta temperature of unhydrolyzed polyacrylamide ( 0 = 265°K (22))... 

Therefore, Flory s theory concludes that as the functionality of a network increases, the constraint contribution, fc, should decrease and eventually vanish. Furthermore, in the extreme limit in which junction fluctuations are totally suppressed, the Flory theory reduces to the affine network model ... 

Further disagreement with the Flory theory is found in the magnitude of 2Cj and 2Cj + 2C2 in Figure 1. These results are quantified In terms of the structure factors A and A2. Aj and A2 relate the small and large-strain moduli to the number of... 

According to the Flory theory (9), the predicted range on Aj and A2 lies between one and (1 —2/). The upper limit, unity, corresponds to affine behavior and the lower limit occurs In phantom behavior. Therefore both A3 and A2 have predicted asymptotes of one at high functionalities and should be Independent of vs/V. 

The values of Aj for the three sets of experiments all display a dependency on Vg/Vj and are well in excess of the limit of unity predicted by the Flory theory (9). The presence of solvent during network formation significantly decreases the values of Aj. Removal of the solvent results in an increase in the values of A. Yet, the values of k for these deswollen networks are still significantly less than those of the networks prepared and tested in bulk. 

It is necessary to notify, that the critical analysis of the Flory theory application for the determination of molecular mass and the crossing density of the coal structure has been done in the Painter s works [16], Authors assert, that the possible formation of hydrogen bonds between the hydroxy groups of low-metamorphized coal has an important role here that is why, even a lot of empirical amendments introduction into calculations leads to obtaining the understated values of molecular masses of clusters. 

Figure 10. Dependence of the reduced equilibrium modulus of POP triol - MDI networks prepared in the presence of diluent. POP triol Mu= 708 stress-strain measurements in the presence of diluent (o) and after evaporation of the diluent ( ). Flory theory for the values of the front factor A indicated, theoretical dependence including trapped interchain constraints Numbers at curves Indicate the value of ry.

In the present article, we focus on the scaled particle theory as the theoretical basis for interpreting the static solution properties of liquid-crystalline polymers. It is a statistical mechanical theory originally proposed to formulate the equation of state of hard sphere fluids [11], and has been applied to obtain approximate analytical expressions for the thermodynamic quantities of solutions of hard (sphero)cylinders [12-16] or wormlike hard spherocylinders [17, 18]. Its superiority to the Onsager theory lies in that it takes higher virial terms into account, and it is distinctive from the Flory theory in that it uses no artificial lattice model. We survey this theory for wormlike hard spherocylinders in Sect. 2, and compare its predictions with typical data of various static solution properties of liquid-crystalline polymers in Sects. 3-5. As is well known, the wormlike chain (or wormlike cylinder) is a simple yet adequate model for describing dilute solution properties of stiff or semiflexible polymers. 

A ternary system consisting of two polymer species of the same kind having different molecular weights and a solvent is the simplest case of polydisperse polymer solutions. Therefore, it is a prototype for investigating polydispersity effects on polymer solution properties. In 1978, Abe and Flory [74] studied theoretically the phase behavior in ternary solutions of rodlike polymers using the Flory lattice theory [3], Subsequently, ternary phase diagrams have been measured for several stiff-chain polymer solution systems, and work [6,17] has been done to improve the Abe-Flory theory. 

The theoretical IA binodals successfully reproduce the experimental bi-nodals for both systems. Furthermore, the theoretical tie lines correctly predict the fractionation effect found by experiment. Thus, the scaled particle theory predicts the IA binodals and tie lines more accurately than the Abe Flory theory. The success owes much to incorporating chain flexibility into the theory. 

We see from the above argument that, within the Flory theory of gels, the concentration dependence of x is the driving force for the transition in neutral gels. Hence, to understand the mechanism of the phase transition of gels on a molecular level, we must identify the microscopic interaction which makes x depend on the concentration. For this purpose, we must specify not only the... 

These phenomenological theories are more complicated than the Flory-type theory, though they have certain advantages over the latter. In the quasichemical treatment, the molecular interaction responsible for the transition, which is hidden behind the parameter % in the Flory theory, appears with clearer physical meaning. In the hole theory of gels, some properties of gels which are... 

These values of g° have been compared with those calculated from Flory theory. 

Since the pioneering work of Kuhn, the solvent freezing point depression observed in swollen crosslinked rubbers has been the subject of many works. The observed AT can be attributed to two origins. A sizable AT is accounted for by the lowering of the thermodynamic potential of solvent molecules in a polymer solution derived from the Flory theory, and the additional AT observed for crosslinked rubbers has been attributed to confinement effects, fn 1991, Jackson and McKenna [32] studied... 

For explicit calculations, one needs to specify the form of the free energy functional 3F pA, PbJ in this self-consistent formulation in detail. Schmid [57] uses for 3 0 an extension of the generalized Flory theory [261]... 

This fact can be demonstrated as follows. Let us determine the value of the well-known Flory parameter x, which corresponds to the 6 point (i.e. to the point of inversion of the second virial coefficient of the solution of rods) in the Flory theory of Ref.9). This can be done by expanding the chemical potential of the solvent in the isotropic phase (Eq. (16) of Ref.9 ) into powers of the polymer volume fraction in the solution, and by equating the coefficient at the quadratic term of this expansion to zero this procedure gives Xe = 1/2 independently of p. On the other hand, it is well known26,27) that the value of x decreases with increasing p and that X < 1 at p > 1. The contradiction obtained shows that the expressions for the thermodynamic functions used in Ref.9) are not always correct... 

It is clear from the above that the conclusions of the Flory theory cannot be regarded as indubitable and require some verification, although they are in a qualitative agreement with experiments. 

Quantitative experimental results in the iield of lyotropic polymeric liquid crystals are now available mainly for solutions of completely stiff macromolecules1,2,4,6. These results are in good qualitative agreement with the conclusions of the Flory theory9 and, thus, with the results in Sect. 2, To reveal the expected deviations from the Flory theory proposed this paper, more accurate and systematic measurements are needed. 

These approximations can then be used in the osmotic equation of state to obtain the compressibility factor. Monte Carlo simulations using the above-discussed Monte Carlo techniques have been performed to assess the approximations inherent in the generalized Flory theory of hard-core chain systems. This theory does quite well in predicting the equations of state of hard-core chains at fluid densities. The question then arises, why does it do so well since the theory typically only incorporates information from a dimer fluid as a reference state ... 

Dickman, R. and Hall, C. K., Equation of state for chain molecules continuous-space analog of Flory theory. J. them. Phys. 85, 4108 115 (1986). 

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