Polymers fitting data

To account for the variation of the dynamics with pressure, the free volume is allowed to compress with P, but differently than the total compressibility of the material [22]. One consequent problem is that fitting data can lead to the unphysical result that the free volume is less compressible than the occupied volume [42]. The CG model has been modified with an additional parameter to describe t(P) [34,35] however, the resulting expression does not accurately fit data obtained at high pressure [41,43,44]. Beyond describing experimental results, the CG fit parameters yield free volumes that are inconsistent with the unoccupied volume deduced from cell models [41]. More generally, a free-volume approach to dynamics is at odds with the experimental result that relaxation in polymers is to a significant degree a thermally activated process [14,15,45]. 

For determining the molar mass of branched polymers gel permeation chromatography can be used. An important quantity in this connection is the hydrodynamic volume of the polymer coil, which, as shown before in Eq. (9.27), is proportional to the product [rj M. According to Benoit and co-workers (1966) the hydrodynamic volume is the key size parameter in the establishment of a universal calibration curve for gel permeation chromatography columns (see Chap. 2) if log (h/]M) is plotted versus the elution volume for a variety of polymers, the data fit a single curve. 

Fig. 2. Fit of the displacement for diffusion with memory to NMR spin-echo measurements in polymers. The data corresponds to the spin-echo signal from polysulfane-poly-butadiene copolymer in different solvents comprising a mixture of trichloroethane (TCE) and octane. The fits were obtained using Eq. (83).

The excluded-volume parameter for PS was derived from viscosimetric data in good solvent collected by Nystrom and Roots for different molecular masses [113], The internal viscosity characteristic time Tq was obtained from best-fit data [12] of mechanical-dynamical results due to Massa, Schrag, Ferry, and Osaki [102,103] (see Figure 10). In analogy with what was previously found by other authors, notably Kirkwood and Riseman on the same polymer (PS) with different solvents [19], fitting the experimental data seems to require two different values of R ff = C/6to/s, one to obtain v q) through Eqn. (3.1.9) and a second one to evaluate to = I kg T, the latter... 

Figure 18.8. I racture toughness vs. area fraction of agglomerates, fit. [Data from Yeh Wang, Jiang-Shen Huang, J. Appl. Polym. Sci., 60, No.l 1, 1996. 1779-91.]...

The modulus formalism is being applied more frequently to fit data of systems largely influenced by conductivity, such as composites [154], semi-crystalline polymers [143], ionic liquids [63] and biological systems [138]. This allows a better resolution of relaxation processes and leads to similar shape parameters and temperature dependence of relaxation times to those achieved by using complex permittivity [143]. 

In deriving equation (12.8) it is assumed that the fractional free volumes of the homopolymers and the copolymers at their respective glass-transition temperatures are the same and that both the specific volume and the occupied volume of the copolymer are obtained by multiplying the corresponding values for the homopolymers by their weight fractions and adding the results. Figure 12.9 shows an example of data for a type of copolymer to which this equation applies, with a= 1.75. The data for many pairs of vinyl polymers fit well to equation (12.8) with a equal to unity. 

TABLE 144 Characteristic Reducing Parameters and the Statistical Fit Data for Polymers and Amorphous Blends... 

Bohdanecky found that published data on many semi-flexible polymers fit eq 3.5. Figure 5-8 shows this with the data [40] for poly (tere-phthalamide-p-benzohydrazide) (PPAH) in dimethyl sulfoxide (DMSO). The linear relation as observed here permits unequivocal estimation of I and S. If we have one more relation among q, Ml, and d, it becomes possible to determine these three unknowns. Molecular weight dependence data of (5 ) or / may be utilized as such an additional relation. Bohdanecky [39] proposed the use of the relation... 

As is known, Tg has been selected as the reference temperature. Equation (4.10) is fitted in the temperature range from to Tg + 100°C. The average values of Cj and Q were obtained by fitting data for a large number of polymers, and estimated to be 17.44 °C and 51.6 °C, respectively, when Tg was chosen as the reference tem-... 

Fig. 16. Monomer concentration vs time trace of a methyl methacrylate/dodecyl acrylate copolymerization at equimolar amounts of both monomers measured in a reaction mixture at 40°C, 100 MPa, and 5 wt% polymer concentration. The difference between measured and fitted data (eq. 73) is illustrated by plotting the residuals (res) in the lower part of the figure. Reprinted with friendly permission by the American Chemical Society.

Molten polymer experimental data were found to fit the relation (see Fig. 11-7) for the NRe range from 5 X 10" to 10". For this range, little effect of viscoelasticity appeared. However, Marshall and Metzner [11], working with polymer solutions, found that (see Fig. 11-8), in a higher range, viscoelastic effects did occur. Since the flow rates encountered in industrial operations would be considerably higher than those of the experimental molten polymer studies [9,10], viscoelastic effects could very well play an important role. 

Several authors note - - that lattice (graph) models (p. 301) offer a comprehensible means of interpreting a large number of properties of polymer solutions with a remarkable success record in furnishing qualitative, if not always quantitative, descriptions. Casassa considers such models open to serious question, but objections have mainly been directed at the lack of molecular interpretation of parameters which are usually obtained by fitting data. 

Experimental data for BA + MMA composites as a function of the polymer composition, w, are compared with the calculated values in Figure 5. Similar results were found for the other two systems (1). Curve fitted data (dashed lines) were close to the theore ical curves (solid lines) for insert A, densities, p ... 

In the first application of the WLF equation, average values of cf and cf were obtained by fitting data on a large number of polymers, and estimated to be 17.44 and 51.6 respectively. It is evident that the actual variation from one polymer to another is too great to permit use of these universal values except as a last resort in the absence of other specific data. In a somewhat better approximation, fixed... 

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