Molar mass dependence

PMDI also contains isocyanates with higher molar masses (triisocyanates, tetraisocyanates, polyisocyanates), whereby the structure and the molar mass depend on the number of phenyl groups. This distribution influences, to a great extent, the reactivity, but also the usual properties like viscosity, flowing and wetting behavior as well as the penetration into the wood surface. 

A plot of logrjsp (Eq. (9)) versus log (c-[r ]) results in a linear relationship. The unknown quantities Bn and n can be obtained from this linear regression (Fig. 7). A correlation of rj0 with concentration and molar mass can now be achieved using a [rj]=KMa relationship and replacing [q] in Eq. (9) by its molar mass dependent form to give ... 

A comparison of the solution behaviour of PS in both solvents, toluene and frans-decalin, reveals that the limiting power of the molar mass dependence of r 0 (3.35 and 3.28, respectively) is very close to the value of 3.4 observed in highly concentrated solutions and melts. The concentration dependence of r 0, however, is clearly different in each of the solvents ... 

Experimental considerations Sample preparation and data evaluation are similar to membrane osmometry. Since there is no lower cut-off as in membrane osmometry, the method is very sensitive to low molar mass impurities like residual solvent and monomers. As a consequence, the method is more suitable for oligomers and short polymers with molar masses up to (M)n 50kg/mol. Today, vapour pressure osmometry faces strong competition from mass spectrometry techniques such as matrix-assisted laser desorption ionisation mass spectrometry (MALDI-MS) [20,21]. Nevertheless, vapour pressure osmometry still has advantages in cases where fragmentation issues or molar mass-dependent desorption and ionization probabilities come into play. 

The molar mass dependence of the intrinsic viscosity of rigid chain polymers cannot be described by a simple scaling relation in the form of Equation (36) with molar mass independent of K and a. over a broad molar mass range. Starting from the worm-like chain model, Bohdanecky proposed [29] the linearizing equation... 

Table 2). All the radii have a certain molar mass dependence. The magnitudes of these radii, however, can deviate strongly from each other. These differences result from the fact that they are physically differently defined. The radius of gyration, R, is solely geometrically defined the thermodynamically equivalent sphere radius, R-p is defined by the domains of interaction between two macromolecules, or in other words, on the excluded volume. The two hydrodynamic radii R and R result from the interaction of the macromolecule with the solvent (where the latter differs from R by the fact that in viscometry the particle is exposed to a shear gradient field).

The interpretation of measured data can be started by examining the molar mass dependence. Here the influence of a broad molar mass dependence has a strong influence, but just this effect can be used for a differentiation between the various mechanisms of their formation and the resulting architectures. 

Generalized ratios of the four differently defined radii can be written. By this manipulation the molar mass dependence is widely eliminated, and the effects of branching becomes more evident. See Table 2. 

Fig. 9. Molar mass dependencies of the intrinsic viscosity of star-branched polyamides obtained by co-condensation of bifunctional amino acids with f-functional polyacids. The curves appear shifted towards smaller intrinsic viscosities as the functionality of the star center was increased [79]. Reprinted with permission from [79]. Copyright [1948] American Society...

Fig. 10. Molar mass dependencies of the radii of gyration for stars of different functionality in a good solvent [25]. From top to bottom, linear, 4,18,32,64 and 128-arms. Reprinted with permission from [25]. Copyright [1993] American Society...

A further remark has to be made when the stars contain polydisperse arms. The radius of gyration is now based on the z-average of the mean square radius of gyration over the molar mass distribution while the degree of polymerization is the weight average Also for this case the molar mass dependence of this radius could be calculated and was found to be [83]... 

Remarkably different molar mass dependencies are obtained with randomly branched or randomly crosshnked macromolecules. Often, below the critical point exponents v in are found which are close to v=0.5, and sometimes... 

These low exponents seem to suggest poor solvent behavior. However, the second virial coefficients are clearly positive and still fairly large and prove good solvent behavior. Surprisingly the molar mass dependencies of and Rj of unfractionated samples are almost indistinguishable from those of their hnear analogues. 

Also the molar mass dependence of the intrinsic viscosity appears odd at first sight. Here exponents in the KMHS equation of a <0.4 are common, and often the exponent decreases further at large molar masses. Figure 16 shows examples. 

Fig. 16. Molar mass dependencies of the intrinsic viscosity [rf] for the same samples as shown in Fig. 15 (end-linked PS-stars [94] and randomly crosslinked polyesters [92,93,95]...

Both effects are the consequence of the difference in averaging Rg and In fact, a fully different picture is obtained when the radii are calculated for the monodisperse fractions. These calculations were first made by Zimm and Stock-mayer [49,97]. Now the expected strong decrease of R with branching was indeed obtained (and also a different molar mass dependence that will be discussed somewhat later). 

The molar mass dependence of the second virial coefficient remains, so far, not fully understood. Why does the exponent in the relationship... 

The starting point was a reconsideration of the molar mass dependence of that is commonly written as... 

The clusters which obey Eq. (61) are self similar to each other. Sometimes, however, the curve flattens at large molar masses and may form another straight line with a different exponent. Such behavior is an indication of a limitation in the separation capability of the column (or some other artifacts) or it is the result of large particles with a different fractal behavior. These particles can be aggregates or clusters of a higher branching density. Similar behavior can be observed also from the molar mass dependence of the viscosity. An example will be shown in the next section. 

Fig. 26. Molar mass dependence of the g factor for three pregel and one postgel fraction of end linked PS stars. A good fit was obtained with the Zimm Stockmayer equation (Eq. 69) and an exponent in Eq. (70) of fi 0.63 [95] which agrees well with Kurata s estimation with b-0.6 [129]. Reprinted with permission from [129]. Copyright [1972] American Society...

A remark had previously been made in various places of the text that the parameter will increase with branching or more precisely with the g-factor. This -dependence must necessarily also result in a molar mass dependence of for which a power law behavior was tentatively assumed. From the KMHS-relation-ship one then finds [95]... 

Thus the molar mass dependence of the CP-parameter can be estimated when, besides the exponent, the fractal dimension of the clusters could be measured. This fractal dimension can be obtained from the molar mass dependence of the radius of gyration of the fractions, or from the angular dependence of the... 

Fig. 27. Molar mass dependence of [rj] for a fractionated comb macromolecule. The fractionation was made with a SEC/LALLS/VISC set-up. The comb macromolecule consists oi a polyimidazole backbone prepared by free radical polymerization. The imidazol side groups acted in a melt with phenylglycidylether and phthalic anhydride as multifunctional initiator for the anionic growth of polyester chains. The straight lines correspond to the behavior of unattached polyester chains and the comb polymers at low and high molar masses respectively [136]...

This section on the contraction factor may be concluded with an example of a comb macromolecule [136]. Due to the route of preparing this comb, unattached side chains also occurred in the system. Figure 27 shows the result of the molar mass dependence of [rj] which was obtained from a SEC/LALLS/VISC fractionation. One observes at low molar masses a straight line with an exponent of 11 =0 JO that coincides with the exponents of linear side chains. It follows a... 

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