Florys Model

Figure A2.5.27. The effective coexistence curve exponent P jj = d In v/d In i for a simple mixture N= 1) as a fimction of the temperature parameter i = t / (1 - t) calculated from crossover theory and compared with the corresponding curve from mean-field theory (i.e. from figure A2.5.15). Reproduced from [30], Povodyrev A A, Anisimov M A and Sengers J V 1999 Crossover Flory model for phase separation in polymer solutions Physica A 264 358, figure 3, by pennission of Elsevier Science.

The Fischer-Tropsch process can be considered as a one-carbon polymerization reaction of a monomer derived from CO. The polymerization affords a distribution of polymer molecular weights that foUows the Anderson-Shulz-Flory model. The distribution is described by a linear relationship between the logarithm of product yield vs carbon number. The objective of much of the development work on the FT synthesis has been to circumvent the theoretical distribution so as to increase the yields of gasoline range hydrocarbons. 

We have recently extended the Flory model to deal with nonpolar, two-solvent, one polymer soltulons (13). We considered sorption of benzene and cyclohexane by polybutadiene. As mentioned earlier, a binary Interaction parameter Is required for each pair of components In the solution. In this Instance, we required Interaction parameters to represent the Interactions benzene/cyclohexane, benzene/polybutadlene, and cyclohexane/ polybutadiene. 

The Suter-Flory model was successfully used to interpret the results of the epimerization reaction carried out on propylene oligomers (204) and on polypropylene itself (106, 205). In both cases a slight prevalence of the r dyad over the m (52/48) is observed. The epimerized polypropylene has a microstmcmre almost coincident with a Bernoulli distribution and represents the polymer sample closest to an ideal atactic polymer so far obtained. 

The Suter-Flory RIS model of PP (A 027 is employed to calculate the 13C NMR chemical shifts expected at the 9-Ca and the CH3 carbons and at the 8- and 10-CHj carbons in the various stereoisomers of the PP model compound 3,5,7,9,11,13,15-heptamethylheptadecane. Differences in the chemical shifts of the same carbon atom in the various stereoisomers are assumed to be attributable solely to stereo-sequence dependent differences in the probability that the given carbon atom is involved in three-bond gauche or y interactions with other carbon atoms. The Suter-Flory model provides an accurate description of the conformational characteristics of PP which permits a detailed understanding of its 3C NMR spectrum. On the other hand, the failure of Provasoti and Ferro s calculations [Macromolecules 1977, 10, 874] is directely attributed to the inadequacies of the Boyd and Breitling three-state RIS model of PP (A 022). 

The Flory model is the version where the equivalence between kinetics and statistical descriptions is extended to the post-gel stage of polymerization. Consequently, the functional groups are assumed to continue to react at random with no distinction on whether they belong to sol molecules or to gel. To analyze this version one can use the explicit form of function H. As usual, the moments are available through successive derivatives of H (Eq. 76) with respect to x calculated at x=l. We may rewrite Eq. (77) in the form... 

The analysis for model 2 of Table 4 (fCy=z/ ) using the Flory model leads to the following moment expressions ... 

If one accepts the above small modification of the Flory model for equilibrium swelling, and also assumes that the turgid end-state is described by the van t Hoff relationship ... 

Flory modeled this by considering the seg-1 ments to be like a swarm of particles tKs- tributed about the center of gravity of the coil in a Gaussian fashion, then considered the balance between the free energy of mixing I and the free energy associated with the elas-1 tic deformation of the chain. He then found an expression for a chain expansion factor, a, that minimized the free energy, which was expressed in terms of a series in the volume fraction of polymer, 0p. The result is shown in Equation 11-58 ... 

Polymer-solvent interactions have been examined by viscometric studies of polymer-solvent-non-solvent mixtures in dilute solution The Fox-Flory model which... 

However, only limited experimental studies on the thermodynamic properties of polypeptide solutions have been carried out. The results of vapor sorption studies for PBLG and poly(P-benzyl L-aspartate) solutions at high polymer concentrations by Flory and Leonard could not be explained by the Flory model, but could be explained by assuming that mixing of solvent with flexible side chains dominates the thermodynamic behavior at high concentrations. Rai and Miller obtained similar results for the PBLG-dimethylformamide (DMF) system at high concentrations. They also showed that the results could be explained by the Wee-MiUer theory in which modification of Flory s lattice theory to allow for side chain... 

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. 

Figure 5. Analysis of MWD of polyethylene sample 1 by 3-component Flory model. Key ...

Low values of the heat of water absorption can be obtained. These values smaller than the liquefaction energy may result from the superposition of an endothermal mechanism breaking of some bonds (4), further cristallization of the polymers (5), expansion of the macromolecules with change of volume. For this last case, which does not correspond to an expansion mechanism described by the Flory model one has to take into account the enthalpy and free energy of expansion (6,7) or the internal pressure due to the polymer (8). 

In this modified Flory model, the liquid crystal-isotropic transition of rods in a solution occurs at... 

Most experimental data fall between the two theoretical expectations with the Flory model representing the upper limit and the Onsager limit representing the lower limit. The two theories are in reasonable agreement with experimental data if the uncertainty of the molecular diameter D and Kuhn length l is taken into account. 

Polymer-solvent interactions have been examined by viscometric studies of polymer-solvent-non-solvent mixtures in dilute solution84 86). The Fox-Flory model which relates the molecular parameters of the unperturbed dimension and the linear expansion coefficient to the total sorption parameter has been used. The latter can be obtained by the simultaneous solution of several Equations when the intrinsic viscosities of the mixtures are known. This method is in an early stage of development and pro-... 

Povodyrev A A, Anisimov M A and Sengers J V 1999 Crossover Flory model for phase separation in polymer solutions Physica A 264 345-69... 

Correlation between the solubility parameter theory SPT and the EoS based on the Flory model, was explored by Biros et al. [1971]. The authors demonstrated that 5 = / V, where P is the... 

The Dill-Flory model may be considered as a more rigorous version of the Hartley model (30). Both models are readily applied to other shapes of micelles, such as rods, discs, bilayers, and vesicles. Also, it follows that diameters of spherical, rodlike, and disclike micelles cannot exceed the total length of two hydrocarbon chains in all-trans conformation. The number of entities in one micelle, i.e. the aggregation number s, is therefore readily estimated for any given chain length r. Assuming equal densities p (= 0.777 g/cm ) for micelles and solid n-alkanes, r may be obtained from the volume v and the constant cross section A (= 2.385 x 10 cm ) of alkane chains ... 

Corrales, L.R., Wheeler, J.C. Tetracritical and novel tricritical points in sulfur solutions A Flory model for polymerization of rings and chains in solvent. J. Chem. Phys. 1989, 90, 5030. 

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