Graessley theory

Ito,Y., Shishido,S. A modified Graessley theory for non-newtonian viscosity of poly-dimethylsiloxanes and their solutions. J. Polymer Sci. Polymer Phys. Ed. 12, 617-628 (1974). 

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. 

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,... 

FIG. 10-20. Logarithmic plot of against 7T /2 for narrow-distribution polystyrene with = 180,000. Different symbols refer to five different temperatures from 142° to 174.5°C. Curve drawn from Graessley theory. (Penwell, Graessley, and Kovacs. ) Reproduced, by permission, from the Journal of Polymer Science. 

The concept of the Graessley theory that the steady-state concentration of entanglements is diminished during flow at high shear rates implies that the steady-state compliance Ry observed in recovery after cessation of steady-state flow will be larger than J and given by the equation ... 

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... 

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." - < ... 

Tw 10 53 weight-average terminal relaxation time, Graessley theory... 

Y. Ito and S. Shishido. Modified Graessley theory for non-Newtonian viscosities of polydimethylsiloxanes and their solutions. J. Polym. Sci. Polym. Phys. Ed., 12 (1971), 617-628. 

Graessley, W. W. Entagled Linear, Branched and Network Polymer Systems. Molecular Theories. Vol. 47, pp. 67—117. 

The empirical frictional factor (T fric) is independent of shear rate but increases in poor solvent this permits to account for the dependence of the scission rate constant on solvent quality. The entanglement part (r enl), as given by Graessley s theory which considers the effect of entanglement and disentanglement processes, is a complex function of shear rate ... 

The bracket (1 — 2/f) was introduced into the theory of rubber elasticity by Graessley [23], following an idea of Duiser and Staverman [28]. Graessley discussed the statistical mechanics of random coil networks, which he had divided into an ensemble of micronetworks. 

The behavior of the small-strain data in the three sets of experiments suggests a more universal format for the small-strain theory of Langley (12) and Graessley (13,16) ... 

Bates (66) has found that, with similar assumptions, Graessley s theory applied to branched polymers leads to rather complicated expressions for the... 

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. 

Graessley s theory, though satisfactory for linear polymers, has not yet been shown to apply to branched polymers. Fujimoto and co-workers (65) attempted to apply it to comb-shaped polystyrenes, but obtained only poor agreement with experiment. They attributed this to the failure of the assumption that the state of entanglement is the same in branched polymers as in linear ones. It is not surprising that this theory fails, for (in common with earlier theories) it predicts that the zero shear-rate viscosity of all branched polymers will be lower than that of linear ones, contrary to experiment. 

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