Stockmayer-Fixman equation

A similar diagram can be derived from the Stockmayer-Fixman equation, which reads in reduced form ... 

Stockmayer-Fixman equation, 252 Storage modulus, 408 Strain, 414 at break, 480 hardening, 814 Stress... 

In the case of the sodium salt of carbox5TnethyIcellulose, Brown and Henley (4S) have reported a series of [rj] values for various pol5uner fractions at different ionic strengths (5.0 10" to 0.2 M NaCl at 25°). These authors show that elaboration of their data according to the Stockmayer-Fixman equation (3) leads to a common K value, independent of ionic strength. Furthermore, from the unperturbed dimension so derived. Brown and Henley conclude that carbox5nnethylcellulose behaves as a "typical flexible polymer in aqueous solution. 

If Equation (9-148) is inserted into (9-153), the Burchard-Stockmayer-Fixman equation (often just known as the Stockmayer-Fixman equation) is obtained ... 

Stockmayer-Fixman Equation If several samples of the same polymer with different molecular weight are available, we can determine the structural parameter K and the interaction parameter X] simultaneously by measuring the intrinsic viscosities. This was suggested by Stockmayer and Fixman (1963) in the equation... 

Among the three equations we have discussed, only the Mark-Houwink equation is exact that is, no assumptions have been made. The Flory-Fox and Stockmayer-Fixman equations are both based on their models. 

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 morphology of methyl, ethyl, cetyl, decyl, and isobutyl copolymers hydrophilic-hydrophobic sequence has been studied and compared in THF and dimethylsulfoxide solvents, using viscosity and light-scattering techniques.The macromolecular chains have a segregated conformation, which allows the solvent to solvate compatible segments of the copolymer. The unperturbed molecular dimension of the isobutyl vinyl ether-MA copolymer are smaller than that for poly (isobutyl vinyl ether), as shown by the Stockmayer-Fixman equation. 

Amu (17) critically reviewed the effect of salts on the solubility of poly(ethylene oxide) in water together with intrinsic viscosity measurements and Flory theta conditions. From this work, Amu estimated the theta temperature of poly(ethylene oxide) in water to be 108.5°C. Amu presented data in terms of the Stockmayer-Fixman equation (18),... 

So a prediction method for the limiting viscosity number can at best give the order of magnitude of this quantity. Such a method will be described on the following pages. The method is based on the empirical relationship between [rj] and M the Mark-Houwink Eq. (9.32). In principle, the Stockmayer-Fixman Eq. (9.38) could be used as well, but the majority of the literature data has been expressed in the constant K and the exponent a of the Mark-Houwink equation. 

Note Calculations have been executed by the following equations 1 - Curata-Yamakawa 2 -Stockmayer-Fixman 3 - Yamakawa-Fudgi. 

By elaboration of their intrinsic viscosity data according to the Stockmayer and Fixman equation, Tanford et al. have arrived at the following average value for the characteristic ratio of randomly coiled proteins (r yjnP 5.0. ... 

The recent theoretical treatment proposed by Miller, Brant and Flory (56) of polypeptide chains containing glydde and aminoacid side chains with branched arbon atoms indicates, as mentioned above that a small number of glycine residues markedly reduces the characteristic ratio, but that a small number of branched side chains does not exert much influence. The figure obtained by Tanford et al., even considering its experimental error of 10% (which is normal), and its origin in the application of the Stockmayer and Fixman equation to low molecular weight samples, is well within theoretical predictions and provides a conclusive test of these. 

Much of the above discussion indicates that, to study excluded volume effects, an accurate determination of unperturbed dimensions is required. For this, a common procedure is to extrapolate intrinsic viscosity of known molecular weight samples to zero molecular weight. Several extrapolations have been used, notably the Stockmayer-Fixman plot. Dondos and Benoit have now introduced a modified version of this, which appears to be linear over a wider range of molecular weights. It introduces a parameter D, which is shown to be linearly related to the exponent of the Mark-Houwink-Sakurada equation. 

Frequently, light scattering experiments are not conducted under theta conditions. This may be because theta conditions are not known or because of the greater ease associated with conducting light scattering measurements in good solvents. Baumann [53] proposed an equation that is equivalent to the Burchard-Stockmayer-Fixman extrapolation procedure for intrinsic viscosity (see below)... 

Besides the simple experimental relationship, Eq. (9.30), between ah and M, quite a number of other equations have been proposed. They always contain a parameter that expresses the interaction between polymer and solvent. For a detailed discussion the reader is referred to the literature. A simple equation between and M has been proposed by Stockmayer and Fixman (1963), which can be written as... 

A large number of methods have been developed for estimating Ke from [tj] measurements in thermodynamically good solvents. These techniques have been periodically reviewed [54,61,62], so only a few key methods will be discussed here. The most widely used method, in part because of its simplicity, is the method of Burchard [63] and Stockmayer and Fixman [64], The BSF equation, which was based on earlier theoretical work by Kurata and Yamakawa [65]... 

A method for correcting intrinsic viscosity data for small departures from the theta state to determine Xg was proposed by Bmchard [ 49 ] and by Stockmayer and Fixman [ 50 ]. While the model on which this method is based is no longer thought to be valid, the equation that results, the BSF equation, has been found useful for dealing with small deviations from theta conditions. The BSF equation is shown here as Eq. 2.91 ... 

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