Viscosity relation, Flory

In summarizing the intrinsic viscosity relations presented in this section, it must be admitted that they represent nothing more than rather small semi-empirical refinements of the Flory excluded volume theory and the Flory-Fox viscosity theory. For a large fraction of the existing body of experimental data, they offer merely a slight improvement in curve-fitting. But for polymers in good solvents it is believed that a more transcendental result has been achieved. The new equations permit more reliable assessment of unperturbed chain dimensions in such cases, and in some instances (e. g., certain cellulose derivatives see Section III B) they offer possible explanations of heretofore paradoxical solution behavior. 

The global rate of the process is r = rj + r2. Of all the authors who studied the whole reaction only Fang et al.15 took into account the changes in dielectric constant and in viscosity and the contribution of hydrolysis. Flory s results fit very well with the relation obtained by integration of the rate equation. However, this relation contains parameters of which apparently only 3 are determined experimentally independent of the kinetic study. The other parameters are adjusted in order to obtain a straight line. Such a method obviously makes the linearization easier. 

The relation (1 10) leads to a number of interesting consequences. In a theta solvent, in which the shape of the chain is described by the random flight model, is proportional to M2, so that the intrinsic viscosity should be proportional to M /2. And this prediction has been applied and verified. In solvent media better than 0-solvents, the theory of Flory [11,46] predicts that the linear expansion factor a increases for any polymer - homologous series with chain length. Thus the exponent v in the empirical equation should be larger than 0.50. 

For a given polymer-solvent system, the intrinsic viscosity varies with the molecular weight of the polymer. According to Flory, the intrinsic viscosity is directly proportional to the hydrodynamic volume occupied by the random coil of the polymer molecule in a solution. In addition, the hydrodynamic volume is related to the cube of the typical linear dimension of the random coil (root mean square end-to-end distance). The intrinsic viscosity is expressed as ... 

In the first case, characteristic viscosity of the fraction, r ], with the known molecular mass in ideal -solvent was measured. This method is based on the known Flory-Fox relation [38], The second method represents measurements of the fraction [r ] in good thermodynamic solvents and extrapolation of experimental data in accordance with the known techniques [39 - 41], Because in the cur-rent work [36] all measurements were performed in good thermodynamic solvent, unperturbed di-mensions of macromolecules, , were determined by graphical extrapolation in accordance with the Shtockmayer-Fixman, suggested by the authors for flexible macromolecules, in [t ] M1/2-M12 coordinates (Figure 6). 

If the polymer is polydisperse, the intrinsic viscosity is obviously related to some kind of average molecular weight, appropriately called the viscosity average by Flory (1943), which depends on the magnitude of o. It is a common practice to define the different averages of molecular weight by the general relationship ... 

Intrinsic viscosity is related to the linear size of the coil R and the molar mass M by the Fox-Flory equation ... 

For example. If the well-known Flory relation (16) Is assumed and the Intrinsic and reduced viscosities are considered comparable at low concentrations, then the solution viscosity may be roughly approximated by... 

The values for xanthan gum were also reported in an earlier work (7). The molecular sizes were obtained by using the Flory relation (8). There are alternate discussions as to what the configuration and the size of xanthan gum molecules in solution are. Whitcomb, Ek, and Macosko have presented an interpretation of the intrinsic viscosity data assuming a cylindrical rod conformation (9). The K-j and a values for Pusher are given by Lynch and Mac-Williams (10). It should be noted that a range of Kj and a values for polyacrylamides can be found in the literature (11). 

Flory hydrodynamic parameter relating intrinsic viscosity to the... 

The distinction between poor and good solvent was introduced in the 1950s by Fox and Flory after experimental studies of the intrinsic viscosity of polymer solutions. These authors recognized that the viscosity varies in relation to the dependence of the chain sizes on temperature the poor solvent state is the state of a solution in which the chains have quasi-Brownian configurations. Systematic experiments have been made in this domain, for instance to determine the Flory temperature, but they have never given very precise results. Physicists are just now beginning to overcome the experimental and theoretical difficulties. Experiments have been made to show the existence of a collapse of the polymer chain, and certain authors have been prone to compare it with the coil-globule transition in proteins. 

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-... 

Based on the Kirkwood-Riseman theory, if polymer chains are nondraining, it follows [Kirkwood and Riseman, 1948 Auer and Gardner, 1955] that intrinsic viscosity data can be related to the radius of gyration, Rg, of flexible polymers. This can be expressed in an equation of the form [Flory and Fox, 1951]... 

Equation (1.70) may be approximated by a simple relation between gr, and g of the form gr, = g. Analysis of experimental intrinsic viscosity data on star polymers in theta solvents indicates that while substantially better than the relation based on the Flory-Fox equation, Eq. (1.70) is still inaccurate, and that a better empirical description is given by [Douglas et al., 1990]... 

The 3.4 power law for melt viscosity, eq 5.1, is of surprising generality, well compared to the Houwink-Mark-Sakurada relation for intrinsic viscosity. Thus its derivation by molecular theory has been one of the central themes in polymer physics since it was proposed in 1951 by Fox tind Flory [63]. Despite many efforts no real success has as yet been achieved. The reader is advised to consult Graessley s review [64] in 1974 concerning the earlier theories. 

The intrinsic viscosity is determined by extrapolating the viscosity to infinite dilution, c 0. In Eq. 2, 7s is the solvent viscosity and A jj is a constant [3]. Thus, the solution viscosity can be written as the sum of the solvent viscosity 7s and the polymer contribution to the viscosity, 7 = 7s + 7p- For an equilibrium coil, is related to the intrinsic viscosity by the Flory-Fox equation, [rj = where... 

Solubility parameters can also be estimated from intrinsic viscosity. Flory [101] related intrinsic viscosity to polymer molecular weight and the chain-expansion factor. The chain-expansion factor can, in turn, be related to the polymer-solvent interaction parameter using the Flory-Hug-gins theory. A variety of models can be used to relate the interaction parameter to solubility parameters [87,102,103] these equations have the form... 

Flory-Fox theory (or Fox-Flory) n. This theory is relates viscosity to molecular dimensions by treating the polymer molecule as a hydrodynamic sphere. In a 0-solvent in which the molecular coU is compact, these authors write the intrinsic viscosity as,... 

PJ. Fox and T.G. Flory developed a theoretical correlation of the intrinsic viscosity [ri]y the radius of gyration Rq and the molar mass M [75]. For the case of hard spheres, the intrinsic viscosity is related to Rq and M if the volume of a polymer coil in Eq. (7.3) is replaced with the volume of an equivalent sphere with a radius of gyration according to Eq. (8.28) ... 

The lattice theory of entropy of mixing, derived independently and contemporaneously by Huggins and Flory and the "Huggins constant K ", relating the concentration dependence of the viscosity of dilute polymer solutions are listed among Huggins major contributions to polymer science. He developed new procedures for more quantitative productions of solubiHty. 

---