Analysis of viscosity data

The T sp and r r values of a dilute solution are represented by a polynomial approximation (neglecting and higher terms in eqn (5.38)) as follows  

Numerous experimental r versus c relations other than Huggins and Kraemer s have been proposed [43]. They are special cases or equivalent to the Huggins equation (eqn (5.45)). 

In very dilute solutions, negative coefficients of d(r sp/ )/dc were observed for some polymer solutions (for example, poly (vinyl chloride) [55] and polystyrene in various solvents [56], and isotactic polypropylene in decalin [57]. 

Interpretations have been given to these extraordinary phenomena, such as adsorption of the polymer molecules on to the surface of the capillary and greater thermodynamic expansion of the polymer molecules in very dilute solution than in the moderately dilute state [58]. 

Kamide (1989). Colligative properties, in Comprehensive Polymer Science, Vol. 1, C. Booth and C. Price (eds), Pergamon Press, Oxford, Chapter 4. 

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The molecular diameters cited above were obtained from an analysis of viscosity data in terms of the Lennard-Jones 6-12 potential ... 

Unfortunately, there are relatively few data for transport coefficients measured under high pressure from which the coefficient (3 In rijdT)v can be calculated. Table I compares the isobaric and isochoric temperature coefficients of viscosity and binary diffusion for a number of liquids. It will be seen that the isochoric temperature coefficients are considerably smaller than those observed at constant pressure. The data of Jobling and Lawrence28 indicate that (d In rj/dT)v is more nearly proportional to (1/T2) than is (d In t]jdT)P. This observation is confirmed in an analysis of viscosity data for a number of hydrocarbons made by Simha, Eirich, and UUmann.31 The small number of substances studied and the shortness of the temperature range covered, however, do not enable us to assume a logarithmic dependence of the isochoric viscosity as a generally valid relationship. 

MATH MODEL ANALYSIS OF VISCOSITY DATA USING THE OSTWALD REGRESSION... 

Figure 5. Math Model Analysis of Viscosity Data Using the Ostwald Regression...

The estimation of f from Stokes law when the bead is similar in size to a solvent molecule represents a dubious application of a classical equation derived for a continuous medium to a molecular phenomenon. The value used for f above could be considerably in error. Hence the real test of whether or not it is justifiable to neglect the second term in Eq. (19) is to be sought in experiment. It should be remarked also that the Kirkwood-Riseman theory, including their theory of viscosity to be discussed below, has been developed on the assumption that the hydrodynamics of the molecule, like its thermodynamic interactions, are equivalent to those of a cloud distribution of independent beads. A better approximation to the actual molecule would consist of a cylinder of roughly uniform cross section bent irregularly into a random, tortuous configuration. The accuracy with which the cloud model represents the behavior of the real polymer chain can be decided at present only from analysis of experimental data. 

Figure 13.20 The rate of thermal degradation of PET as a function of the intrinsic viscosity measured at different temperatures 1, 280 °C 2, 290°C 3, 300°C 4, 310°C [29b, 29c, 39], From Thermal degradation of PET. A kinetic analysis of gravimetric data, Covney, J. D., Day, M. and Wiles, D. M., J. Appl. Polym. Sci., 28, 2887 (1983), copyright (1983 John Wiley Sons, Inc.). Reprinted by permission of John Wiley Sons, Inc.

There are a number of techniques that are used to measure polymer viscosity. For extrusion processes, capillary rheometers and cone and plate rheometers are the most commonly used devices. Both devices allow the rheologist to simultaneously measure the shear rate and the shear stress so that the viscosity may he calculated. These instruments and the analysis of the data are presented in the next sections. Only the minimum necessary mathematical development will he presented. The mathematical derivations are provided in Appendix A3. A more complete development of all pertinent rheological measurement functions for these rheometers are found elsewhere [9]. 

Dendrimers have a star-like centre (functionality e.g. 4) in contrast to a star however, the ends of the polymer chains emerging from the centre again carry multifunctional centres that allow for a bifurcation into a new generation of chains. Multiple repetition of this sequence describes dendrimers of increasing generation number g. The dynamics of such objects has been addressed by Chen and Cai [305] using a semi-analytical treatment. They treat diffusion coefficients, intrinsic viscosities and the spectrum of internal modes. However, no expression for S(Q,t) was given, therefore, up to now the analysis of NSE data has stayed on a more elementary level. 

Precision and Accuracy. Table I shows the results of tests on the linear polyethylene NBS Standard Reference Material No. 1475. This reference material has a pellet-to-pellet coefficient of variation of 3% in the limiting viscosity number according to the NBS Certificate. At least 50 pellets are recommended for a representative sample on which limiting viscosity number is obtained. Nine analyses were performed over a period of about one month on the pellets, using approximately three pellets per determination. The analysis of these data is shown in Table I along with a comparison with the NBS data and their estimate of precision. 

Intrinsic Viscosity—Molecular Weight Relationship for PMMA in TFE. The intrinsic viscosities of the PMMA preparative GPC fractions and whole polymers in TFE at 50 °C and in benzene at 30 °C are shown in Table III and plotted in Figure 4. A least-squares analysis of the data plotted in Figure 4 yields the relation... 

The ZDDP deterioration By reference to Table 6.10 and the case study 2 Evaluation of ZDDPs , in field tests of 56 passenger car vehicles (taxi cabs) and laboratory analysis of lubricating data were included viscosity, TBN, TAN, ZDDP (active), dispersants, oil consumption rate, engine deposits, camshaft and valve lifter wear. Which of the major ZDDPs and ZDDPs mixtures provide the best antiwear and antioxidant performance ... 

In summary, it has to be mentioned that in many studies intrinsic viscosity, inherent viscosity and dilute solution viscosity (DSV) were used in order to monitor the increase of molar mass on monomer conversion. Unfortunately, only a few studies use GPC rather than viscosity measurements. For a few Nd-carboxylate-based catalyst systems linear dependencies of Mn on monomer conversion were established and proof in favor of requirement No. 2 linear increase of Mn with monomer conversion (no irreversible chain transfer) was provided. A more detailed analysis of the data, however, reveals deviations from linearity particularly at low monomer conversions (< 20%). These deviations are particularly pronounced for polymerizations with induction periods. Also the extrapolation of the straight lines to zero monomer conversion reveals intercepts on the Mn-axis. 

The product X,oTi is known as the Walden product and is the focus of the discussion in the present section. To a first approximation, A,oT1 should be independent of solvent nature if the ion moves in the solution without any accompanying solvent molecules. However, some variation in A,oT1 is seen on the basis of an analysis of the data in tables 6.4 and 6.5 using the viscosities reported in table 6.1[13]. 

Care should be taken when analysing the rheological results as coalescence leads to an increase in droplet size that is usually followed by a reduction in the viscosity of the emulsion. This trend is only observed if the coalescence is not accompanied by flocculation of the emulsion droplets (which results in an increase in the viscosity). Ostwald ripening can also complicate the analysis of rheological data. 

A least-square analysis of the data in the lower pressure range results in a AV value of — 9.6cm3mol whereas the data in the higher pressure range results in a value of +7.1 cm1 mol" These results clearly demonstrate how the different steps in Reaction (50) can become rate limiting as a function of the viscosity of the medium and the applied pressure. 

Finally, a close analysis of the data shows that the molecules in the ordered solutions still possess considerable freedom of orientation, as predicted by theory (15). This follows in the first place from the complete data of viscosity as a function of shear stress and concentration (9), which show a large decrease of viscosity with the shear stress in the ordered solutions, in the same range where the viscosity of the random solutions decreases with the shear stress because of the ordering of the molecules paral-... 

In Figure 7.7, the plots of r vs. < > calculated from Simha s Eq 7.24, Mooney s Eq 7.28, and Krieger-Dougherty s Eq 7.8 are compared with the empirical curve-htted relation, Eq 7.5. For all the relations, the intrinsic viscosity [t]] = 2.5 was used. However, to optimize the fit, different values for the maximum packing volume fraction, ( ) = 0.78, 0.91, and 0.62 respectively, had to be used. Detailed analysis of Thomas data made it possible to conclude... 

A viscosity-molecular mass relationship [197] for one of the polynaphthoyl-enebenzimidazoles is presented in Table 28. Some characteristics of the synthesized polynaphthoylenebenzimidazoles containing 1,1-dihalogenethyIene groups are tabulated in Table 29. From analysis of the data, all the synthesized polynaphthoylenebenzimidazoles are soluble not only in sulfuric acid but in phenolic solvents as well. The majority of the systems are characterized by the high softening temperatures and high decomposition temperatures under the conditions of dynamic TGA,... 

The conductivities of MCIO4 (M = Na, K, Rb, or Cs) in ethylene glycol have been determined and the temperature coefficients of their mobilities estimated the analysis of the data shows that the ions are strongly solvated. Observations noted in studies of the viscosities of solutions of MI (M = Li, Na, K, Rb, or Cs) in DMSO also indicate that solvation of M is important in this solvent. ... 

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