Flory-Fox parameter

Equation (3.60) has just the form of the well-known Flory-Fox equation (98). As is well-known, this equation holds for sufficiently high molecular weights. With decreasing molecular weights shielding is expected to become less effective [cf. also Kirkwood and Riseman ( 8)]. Apparently, the Flory-Fox parameter of eq. (3.60) reads... 

At this point it seems of interest that also Fixman (107) used such a scaled Gaussian distribution. This author treated the influence of the excluded volume on the intrinsic viscosity, using the subchain model, as well. However, in this work the influence of the excluded volume is treated more correctly. The Gaussian distribution is only scaled in order to make the correction terms due to the excluded volume potential as small as possible. Fixman arrived at the following lower limit of the Flory-Fox parameter ... 

In ref. 63 a lower numerical factor than that of eq. (5.3) is given. This is due to the fact that the original Flory-Fox parameter, viz. 2.1 X 1023, has been used there (98). The present numerical factor is preferred as it is consistent with the concept of Gaussian coils in the absence of excluded volume. A comparison between experimental results of ref. (69)... 

The right-hand side includes only the average over the equilibrium distribution function. It can be shown that eqn (4.187) gives a correct viscosity for two limiting cases, i.e., spherical particles and rigid rodlike particles. For flexible polymers in the 0 condition, however, the bound is rather weak in terms of the Flory-Fox parameter, one obtains (after some tedious calculation) ... 

The constant 0, called the Flory-Fox viscosity parameter, has an experimental value of x2.5 x 1023 mol-1 [44]. The corresponding value for x2 is then ... 

The thermodynamic linear expansion factor has been related to Flory or thermodynamic interaction parameter, %, and the entropy of dilution parameter, Xs, through the Flory-Fox [10] equations. 

In GPC, the product [77] M, (or the hydrodynamic radius Re) has been widely accepted as a universal calibration parameter. In the Ptitsyn-Eizner modification of the Flory-Fox equation the quantity 4>, which relates the dimensional parameters to the above product, is taken as a variable. The value of < depends upon molecular expansion in solution as represented by a function f(e). Because of this dependence polymeric species having the same [77] M value cannot have the same statistical dimensions (radius of gyration or end-to-end distance) unless they have the same e value. Thus, if [77] M is a universal calibration parameter, the statistical parameters cannot be used as such. A method is presented for obtaining the Mw/Mn ratio from GPC data even though universal calibration is used. 

The ratio (Re/R)3 = (I)3 is thus implicit in the value of in the Flory-Fox equation and has a value of 0.49, corresponding to the Flory-Fox value of 2.1 X 1023. It is clear from Equations 1, 2, and 3 that [77] M cannot be related to the statistical polymer dimensions h and R without a knowledge of , i.e., < , which varies with solvent for a given polymer. It follows, that if all species having the same [77] M elute together from the GPC columns, then only Re can be the universal parameter, since will not be the same for all solute-solvent pairs and h and R will not be equally correct for universal calibration. 

Unfortunately, even the universal calibration does not approach the real molar mass value, although it is often misinterpreted as an absolute molar mass determination method. The reason for this diversity is that the intrinsic viscosity is incorporated into this calculation as a measure for the spatial dimensions of the molecules. Taking into account the influence of the molecular volume, according to the Flory-Fox equation (Equation 24.6), this should lead to a real molar mass value if the universal parameter is constant and equal for all polymers. However, as shown below, this is not the case for the hb polymers. 

I he interaction x, and flexibility parameters are determined from [ ) and M measured on a series of polymer fractions in a good solvent by means of Flory-Fox (1951) equation. The latter results from substitution of a from Equation 72 into Equation 3.1-113... 

For example, in Chapter 6, to begin with three parameters, p (shear stress), e (shear strain), and E (modulus or rigidity), are introduced to define viscosity and viscoelasticity. With respect to viscosity, after the definition of Newtonian viscosity is given, a detailed description of the capillary viscometer to measure the quantity t follows. Theories that interpret viscosity behavior are then presented in three different categories. The first category is concerned with the treatment of experimental data. This includes the Mark-Houwink equation, which is used to calculate the molecular weight, the Flory-Fox equation, which is used to estimate thermodynamic quantities, and the Stockmayer-Fixman equation, which is used to... 

The chemical composition, dimensions and the molecular parameters of four dental acrylic polymer materials and two polyacrylic acids of different molec.wt. used as model compounds were studied by NMR and IR spectroscopies, GPC, intrinsic viscosity and light scattering measurements. Two of the dental polymers were shown to be mainly polyacrylic acid, while one showed the peak pattern for an acrylic acid/methyl acrylate copolymer of about 2 1 composition and one was an oligomer derived from 2-hydroxyethyl methacrylate. The steric hindrance parameter, molecular stiffness and second virial coefficient were calculated using different thermodynamic models. The Flory-Fox-Shafgagen and the Stockmayer-Fixman models fitted the experimental data reasonably well and could be use to describe the molecular parameters of the acrylic polymers. 18 refs. (7th International Conference on Polymer Characterization (POLYCHAR-7), Denton, Texas, USA, Jan. 1999)... 

It is clear that the Flory-temperature is the critical miscibility temperature in the limit of infinite molar weight. Fox (1962) succeeded in correlating 0F-temperatures of polymer-solvent systems with the solubility parameter 5S of the solvent. Plots of 8S as a function of 0F are shown in Fig. 7.8. 

Table 6.3. Data [100-118] gathered to develop a relationship for the parameter Kg of the equation of Fox and Flory, the polymer property (Tgoo) used to fit the Kg values, and the fitted Kg values, for 35 polymers. Tgoo values were rounded off to the nearest integer and Kg values were rounded off to the nearest thousand for simplicity in this table, but not in the dataset analyzed in order to develop the relationship. TgCo is in degrees Kelvin, while Kg is K-g/mole.

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-... 

Figure 35.7 Plots of ulc versus for polyvinyl acetates in benzene at 20°C. Data are from C. R. Masson and H. W. Melville, Poly. 5c/., 4, 337 (1949). Curves are drawn using Eq. (35.20) with Eg = values for (tc/c )q and were calculated from parameters given by T. G. Fox, Jr., P. J. Flory, and A. M. Bueche, JA.C.S. 73, 285 (1951). (Units for T, are m /kg.)...

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