Graessley ratios

Graessley and his co-workers have made calculations of the effects of branching in batch polymerizations, with particular reference to vinyl acetate polymerization, and have considered the influence of reactor type on the breadth of the MWD (89, 91, 95, 96). Use was made of the Bamford and Tompa (93) method of moments to obtain the ratio MJMn, and in some cases the MWD by the Laguerre function procedure. It was found (89,91) that narrower distributions are produced in batch (or the equivalent plug-flow) systems than in continuous systems with mixing, a result referrable to the wide distribution of residence times in the latter. 

In principle, intrinsic viscosities used for estimating branching should be measured under conditions where the expansion factor a is unity, but as indicated in Section 6, it is not easy to identify such conditions. Some authors, e.g. Moore and Millns (40) have measured [tf at the theta-temperature of the corresponding linear polymer, but it is doubtful whether a is unity at that temperature for either linear or branched polymer, if the theories of Casassa or of Candau et al. are valid. If a were the same for both linear and branched polymers under the same conditions g would be unaffected and g could be measured at any convenient temperature some authors have presented data suggesting that g is nearly the same in good and poor solvents, e.g. Hama (42) and Graessley (477), but other authors, e.g. Berry (43) have found g to vary. The best that can be done at present would appear to be to measure g at the theta-temperature on the assumption that this ratio will be less temperature-sensitive than either intrinsic viscosity, and that even if this temperature is not the correct one it will be near it. Errors in estimates of branching due to this effect are likely to be much less serious than those due to the use of an incorrect relation between g and g0. 

Fig. 12.12 Comparison of the viscosity and swelling ratio dependence on shear rate for a polystyrene melt of Mw = 2.2 x 105 and Mw/Mn = 3.1. [Reprinted hy permission from W. W. Graessley, S. D. Glasscock, and R. L. Crawley, Die Swell in Molten Polymers, Trans. Soc. Rheol., 14, 519 (1970).]...

For the weakly entangled system, the steady-state modulus depends on the molecular weight of polymer as M 1, while for strongly entangled system, the steady-state modulus does not depend on the molecular weight of polymer, which is consistent with typical experimental data for concentrated polymer systems (Graessley 1974). The expression for the modulus is exactly the same as for the plateau value of the dynamic modulus (equations (6.52) and (6.58)) Expressions (9.42) lead to the following relation for the ratio of the normal stresses differences... 

FIG. 16.18 Non-Newtonian viscosity ratio for solutions of narrow molecular weight distribution polystyrenes in n-butyl benzene, plotted vs. reduced shear rate q/q0, where qa, equal to the reciprocal of the characteristic time constant Tn, is chosen empirically for each solution. The data were obtained for molecular weights ranging from 160 to 2400 kg/mol and for concentrations ranging from 0.255 to 0.55 g/ml at temperatures from 30 to 60 °C.The full line is calculated with the aid of Eqs. (16.52)—(16.55). From Graessley, Hazleton and Lindeman (1967). Courtesy Society of Rheology. 

Figure 10.3 J>-3 Dispersion ratio versus con-versUm, calculated for the three reactor types with typical parameter values for vinyl acetate polymerization (from Nagasidtra-manian and Graessley [26]).

The molecular theory of rubberlike elasticity predicts that the first coefficient, Ci, is proportional to the number N of molecular strands that make up the three-dimensional network. The second coefficient, C, appears to reflect physical restraints on molecular strands like those represented in the tube model (Graessley, 2004) and is in principle amenable to calculation. The third parameter,, is not really independent. When the strands are long and flexible, it will be given approximately by 3X, where Xm is the maximum stretch ratio of an average strand. But is inversely proportional to N for strands that are randomly arranged in the unstretched state (Treloar, 1975). Jm is therefore expected to be inversely proportional to Ci. Thus the entire range of elastic behavior arises from only two fundamental molecular parameters. 

In the case of good solvent power, x, is approximately given by the Rouse relaxation time which corroborates the basic assumptions of the Graessley theory. However, in a poor solvent and in the vicinity of demixing the ratio X(,/x increases which clearly demonstrates the reduced mobility of the polymer chains under these circumstances. A similar feature can be observed in the course of the study of the poly(n-methyl methacrylate)s Figure 11 shows the shift factor Xq divided by the corresponding quantity obtained at the same concentration in a good solvent. 

I CR time Xcr for a sequence of Ncr of PtBS in the blend. This CR term is expressed in the discrete Rouse form with the eigenvalue ratio given by Equation (3.72). Because the local CR hopping of the PtBS chain is activated by the global motion of the PI chains, the onset time for the CR process, Xcr /CjNcR-ir should be determined by x pi of PI. Watanabe et al. (2011) utilized the Graessley model (Graessley, 1982) to relate xcr / /ncr-i and x h as... 

The molecular theories of Bueche and Graessley are similar in that both theories relate the pseudoplastic nature of polymer solutions as a function of a dimensionless Deborah number which represents a ratio of the response time of the polymer molecules in solution, X, to the time scale of the flow process (18). In simple shearing the time scale of the flow process is inversely proportional to the shear rate. Thus both equation (5), developed from the Bueche theory, and equation (14), developed from the Graessley theory, can be expressed in terms of the Deborah number,... 

For polymers with sharp molecular weight distribution, a terminal relaxation time T] can usually be determined experimentally from the flnal stages of stress relaxation either after sudden strain or after cessation of steady-state flow the latter kind of experiment weights the desired parameter more strongly as can be shown by equations 19 and 64 of Chapter 3, when expressed in terms of a discontinuous set of relaxation times rather than a continuous spectrum. Alternatively, it can be obtained from the constant Ag (the ratio of G /(tP at very low frequencies). Since the very narrow distribution of relaxation times in the terminal zone is close to a single terminal time t, which may be approximately identified with t , of the Graessley theory or of the Doi-Edwards theory (Section C3 of Chapter 10), equation 3 of Chapter 3 applies approximately and... 

For M M c, if there were really a single terminal relaxation time as implied by equation 14, combination of that equation with equation 34 of Chapter 3 would make the steady-state compliance the same as the plateau compliance 7 = 7/v = /G%. with J% given by equation 2. Actually, as shown by Graessley, the ratio Je/J% is the ratio of what may be termed the weight- and number-average relaxation times in the terminal zone ... 

FIG. 17-25. Non-Newtonian viscosity ratio t]/rjo for solutions of narrow-distribution polystyrenes in fl-butyl benzene, plotted logarithmically against log yTt, with characteristic time constant t, chosen empirically for each solution. Molecular weights from 0.16 to 2.4 X 10. concentrations from 0.20 to 0 55 g/cc. Solid curve from Graessley theory." - < ... 

Shroff and Mavridis [56] examined the simpler problem of inferring a single polydispersity parameter from various types of rheological data. They considered several parameters in addition to the polydispersity index MJMJ, including the polydispersity index of relaxation times originally defined by Graessley [57]. This is the ratio of average relaxation times defined by Eqs. 4.56 and 4.57. 

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