Viscosity molar-mass dependence

Figure 2.4. Typical viscosity-molar-mass dependence for molten polymers. From Macosko (1994, Figure 11.5.4). Cop)right (1994). Reprinted with permission of John Wiley and Sons, Inc.

Mendichi, R., Soltes, L., and Schieroni, A. G., Evaluation of radius of gyration and intrinsic viscosity molar mass dependence and stiffness of hyaluronan. Biomacromolecules, 4,1805-1810 (2003). 

Another method of avoiding the time-consuming individual measurements of different molar mass samples is the online coupling of SEC with viscosity detector this provides information on the intrinsic viscosity/molar mass dependence, within... 

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. 

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... 

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...

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]...

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. 

Equation (8.136) is tested in Fig. 8.17 (solid curves) and found to describe the molar mass dependence of constant friction coefficient viscosity data for all three of these linear polymers. The critical molar mass Me for entanglement effects in viscosity is always a factor of 2-4 larger than the entanglement molar mass Mg. that was defined in Eq. (7.47). 

The simple reptation model does not properly account for all the relaxation modes of a chain confined in a tube. This manifests itself in all measures of terminal dynamics, as the longest relaxation time, diffusion coefficient and viscosity all have stronger molar mass dependences than the reptation model predicts. Tn Sections 9.4.5 and 9.6.2, more accurate ana-... 

Molar mass dependence of viscosity for polymer melts reduced by their critical molar mass. Open circles are polyisobutylene with M = MnnOgmor, from T. G. Fox... 

Recall that Fig. 9.3 showed the linear viscoelastic response of a polybutadiene melt with MjM = 68. The squared term in brackets in Eq. (9.82) is the tube length fluctuation correction to the reptation time. With /i = 1.0 and NjN = 68, this correction is is 0.77. Hence, the Doi fluctuation model makes a very subtle correction to the terminal relaxation time of a typical linear polymer melt. However, this subtle correction imparts stronger molar mass dependences for relaxation time, diffusion coefficient, and viscosity. 

The probability distribution function of the tube length L for a chain with N monomers is approximately Gaussian, with mean-square fluctuation of the order of the mean-square end-to-end distance of the chain. The tube length fluctuates in time, leading to stronger molar mass dependences of relaxation time, viscosity, and diffusion coefficient resembling experi-mental observations over some range of molar masses ... 

Figure 9-26. A plot according to Equation (9-151) of the molar mass dependence of the intrinsic viscosity of poly(7-l>enzyl-L-glutamate) in dichloroacetic acid (DCA) and dimethyl formamide (DMF) at 25°C. Coils occur in DCA and helices occur in DMF (according to P. Rohrer and H.-G. Elias).

A comparison of the expressions giving the molar mass dependence of the intrinsic viscosity shows that the relationships can be generalized for all macromolecular types in the form of the modified Staudinger equation [77] = KM (see also Table 9-7). Both K and a, are usually unknown. For each polymer homologous series, therefore, the modified Staudinger equation must be empirically determined. To do this, the molar masses and intrinsic... 

The critical molar mass (Me) parameter [18,19] denotes the transition in the melt viscosity/molar mass relation as the exponents change from 1 to 3.4. Table 25.5 presents the polymers for which the Me values are known while Table 25.6 lists the polymers for which, seemingly, Mc/Mg is one. The ratio of Me/Mg was long taken to be 2 [18] and thus to be species independent. However, a recent empirical compilation [19] has shown that the ratio is p dependent and varies from 3.5 (PE p = 1.69) to 1.4 (a-PCHE p = 5.59). Based upon the data of Table 25.5 this ratio is empirically expressed as ... 

The molar mass determined with an [/jJ-M-relationship is the so-called viscosity average molar masses M, rather than mass (M ) or number average (MJ molar mass (see Chap. 2). This seems to be peculiar on first glance, since the molar masses used for the determination of the [ /]-M-relationships were either number or mass average molar masses (depending on the determination method). However, the samples used for the determination of the [/j]-M-relationship should have had a very narrow molar mass distribution. The unknown sample on the other hand has an unknown distribution function of the molar mass. Only in the case that the distribution function of the unknown sample and the sample used for the determination are the same, or M are determined directly from the viscosimetric measurement. 

Vallerien et al. [126] were the first to describe the Goldstone mode and soft mode of the FLCPs. Due to the higher molar weight of polymer liquid crystals, the rotational viscosity increases and the relaxation frequencies are moved to lower values (usually two or three orders of magnitude). This behavior was also described by Pfeiffer et al. [127]. The authors show the molar mass dependence of soft mode relaxation frequencies at the transition Sm C /Sm A (Fig. 35) and compare characteristic switching times, relaxation frequencies of the Goldstone mode, and the... 

A lot of molar mass dependencies can be described by eq. (4.1.4). Most of the thermodynamical properties of polymers, e.g. T, Tg, depend on M . Often, the limiting value is attained aheady at molar masses below the real molar mass range of a polymer. Then, no dependence of the characteristics can be observed from the molar mass. Properties of bulky polymers connected with large deformations, e.g. melt viscosity, solution viscosity, are determined by M . Viscoelastic properties, e.g. melt elasticity, depend on M. ... 

The Rouse theory is clearly not applicable to polymer melts of a molar mass greater than (M ) for which chain entanglement plays an important role. This is obvious from a comparison of eqs (6.40)-(6.42) and experimental data (Figs 6.13 and 6.14) and from the basic assumptions made. However, for unentangled melts, i.e. melts of a molar mass less than (M ), both the zero-shear-rate viscosity and recoverable shear compliance have the same molar mass dependence as was found experimentally (Figs 6.13 and 6.14). The Rouse model does not predict any shear-rate dependence of the shear viscosity, in contradiction to experimental data. 

In order to clarify the fractal nature of hyperbranched polymers, we first fractionated the polydisperse perfect hyperbranched polystyrene homopolymers into a series of narrowly distributed samples, and determine their molar mass-dependent molecular parameters by a combination of LLS and viscosity measurements. 

In contrast, all the hyperbranched polystyrenes used in the current study were fractionated by precipitation, i.e., according to the interaction parameter xM, where M is the overall molar mass of the chain and x is the Flory-Huggins parameter, a constant, independent on M for a polymer in a given solvent. Therefore, hyperbranched chains in each fraction used here have a similar molar mass. In comparison with our results, some of previous computer simulations [27, 28] suggested that the molar mass dependence of intrinsic viscosity of irregular hyperbranched chains deviates from Eq. 5.2 because the segment density distribution is different from those predicted by de Gennes and Hervet. We will come back to this point later. 

The table gives the UCST and LCST values for selected polymer/solvent pairs at specified values of the molar mass, as measured by the number average M, mass average or viscosity average M. Further information on the molar mass dependence of UCST and LCST can be found in the references. 

Fig. 6.14. Molar mass dependence of the zero shear rate viscosity of PI. The lower figure indicates for Mw > 2 x 10 gmon a power law rjo oc M. The critical molar mass as determined by the location of the break is = 1.3 x 10 gmol . Prom Pyckhout-Hintzen et al. [72]...

Gpi denotes the plateau modulus and Tq, is the mean relaxation time of the Rouse mode part, agreeing with the mean relaxation time of a Rouse system of chains with Nr equal to F7r,c- Because the first term on the right-hand side of Eq. (8.117) is constant, for the molar mass dependence of the viscosity of entangled melts we obtain the expression... 

The determination of the intrinsic viscosity of the polymer in the solution showed that the concentration where the light scattering data were taken is far above the overlap concentration hence there is no appreciable molar mass dependence. 

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