Molecular weight series expansions

In Vienna, Mark published a number of fundamental papers. Their topics include polymerization mechanism (46, 47, 48), thermal polymerization (49, 50), polymerization kinetics (51), the effect of oxygen on polymerization (52), and measurement of molecular weight distribution (53). Guth and Mark expanded their modeling of extended and balled thread molecules to include rubber. The result of their studies was a series of very important papers in which the thermal effect on expansion and relaxation of rubber is explained (54, 55, 56). 

Morphological structures and properties of a series of poly(ethyl acrylate)/clay nanocomposites prepared by the two distinctively different techniques of in situ ATRP and solution blending were studied by Datta et al. [79]. Tailor-made PNCs with predictable molecular weights and narrow polydispersity indices were prepared at different clay loadings. WAXD and studies revealed that the in situ approach is the better option because it provided an exfoliated morphology. By contrast, conventional solution blending led only to interlayer expansion of the clay gallery. 

The Eqs. (B.15) and (B.19) are the basic relationships for the calculation of static conformational properties. Having solved the various sums occuring in these relationships, the correct and apparent molecular weight is obtained by setting q = 0, and the mean-square radii of gyration are the corresponding first coefficients in the series expansion of MwPz(q) and MwPapp (q), respectively... 

Where R is the gas constant, T the absolute temperature, and M the molecular weight of the polymer. This series is usually called the osmotic virial expansion, with A (i = 2, 3,...) being referred to as the i-th virial coefficient of the... 

When dwj fdi = 0, the curve has a maximum and / = I / In p. A series expansion of — 1 / In p gives p/(p — I) as the first term and this fraction approaches I /(p — I) as p approaches 1. This is the value of in linear step-growth polymerizations (Eq. 5-20). Thus, as a first approximation, the peak in the weight distribution of high conversion linear step-growth polymers is located at of the polymer if the synthesis was carried out under conditions where interchange reactions and molecular weight equilibration could occur. 

The smooth change in the polymer coil dimension passing through the 6-temperature is exemplified by the data of Pritchard and Caroline (1981) shown in Fig. 6.4. Here the hydrodynamic expansion factor is plotted as a fimction of the temperature. Contrary to the predictions of the blob theory, da /dr is clearly non-zero at the 0i -temperature, irrespective of the molecular weight. Its positive value is consonant with the predictions of the cluster series... 

In the general case, x should be, of course, a function of the polymer concentration and molecular weight (more rigorously, a function of the molecular-weight distribution). The concentration dependence of x can be represented, generally, by means of an expansion series in terms of uj... 

The entire set of coupled, simultaneous equations was solved numerically, employing the method of characteristics. The particle size distributions were generated directly by the numerical printout from the computer. Molecular weight distributions were obtained by using the calculated moments in a general distribution function written in terms of a series expansion of the individual moments (1). 

Koenderink, et al. examined the motion of perfluorinated hydrocarbon spheres through xanthan solutions(72). Depolarized QELSS spectra were measured at a series of angles and fitted to second-order cumulant expansions. The spheres had radius 92.5 nm the xanthan molecular weight was 4 MDa. Koenderink, et al. measured solution viscosity, shear thinning, storage and loss moduli, translational and rotational diffusion coefficients Dp and Dr of the probes, and probe sedimentation coefficient s, and made an extensive and systematic comparison... 

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