The Steady-State Compliance

The steady-state compliance /° of a linear, monodisperse polymer increases linearly with M when M is less than a critical value M q but becomes independent of M when M M q, i.e., when entanglements become important. Curiously, M q is much larger than Mq, the critical value of M for the effect of entanglements on 7o, often by a factor of four or five. This behavior is sketched in Fig. 5.12. 

An explanation of the dependence of on M when M M q according to the Rouse-Beuche theory of imentangled melts is given in Chapter 6. The equations governing behavior in the two regimes are as follows  

For polydisperse materials the steady-state compliance is very sensitive to molecular weight distribution. This effect shows up even in so-called monodisperse samples. Fuchs ef al. [38] fitted the following empirical equation to their data for a series of PMMAs having polydispersity indices (M /M ) less than 1.15. 

For more non-uniform samples the effect is much more dramatic. For example, in a blend of two compatible, linear, monodisperse polymers /° can be several times larger than the values of 7° of either of the two components. This is illustrated in Fig. 5.13, showing the steady-state compliances of two monodisperse samples as well as those of various binary blends of 

Because of its strong dependence on polydispersity even a tiny amount of high-molecular weight polymer can increase significantly, and this can cause problems in the experimental 

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Chain branching affects the viscosity, the longest relaxation time, and the steady-state compliance and therefore influences creep and stress relaxation (19,163- 167). The effect is difficult to quantify because the length... 

Star-shaped polymer molecules with long branches not only increase the viscosity in the molten state and the steady-state compliance, but the star polymers also decrease the rate of stress relaxation (and creep) compared to a linear polymer (169). The decrease in creep and relaxation rate of star-shaped molecules can be due to extra entanglements because of the many long branches, or the effect can be due to the suppression of reptation of the branches. Linear polymers can reptate, but the bulky center of the star and the different directions of the branch chains from the center make reptation difficult. 

Now as t -> 0 the spectral function L reduces to the area under the distribution. This is the steady state compliance Je ... 

The greater melt viscosities observed for some branched polymers, as compared with linear ones of the same MW, are not accounted for by current theories, as indicated in Section 5. The greater values of the steady state compliance mentioned above is also unexpected theory (128) would suggest a difference in the opposite sense. 

Portions of the literature on viscoelasticity in concentrated polymer systems of narrow distribution have been reviewed recently (15, 16, 152, 153). The following discussion concerns three principal characteristics, the viscosity-molecular weight relation, the plateau modulus, and the steady-state compliance. 

The steady-state compliance data have been analyzed in reduced form... 

Holmes, L. A., Ferry, J.D. Dependence of the steady-state compliance on concentration and molecular weight in polymer solutions. J. Polymer ScL Pt C 23,291-299 (1968). 

Kusamizu,S., Holmes,L.A., Moore,A.A., Ferry,J.D. The steady-state compliance of polymer solutions. Trans, Soc. RheoL 12,559-571 (1968). 

From the results of Section 3.7 it becomes obvious that extremely sharp fractions are needed for a check of eqs. (3.41a) or (3.42). Only in this case a quantative agreement of the experimental values of the steady-state compliance JeR with the theoretical ones, as given by the eqs. (3.61), (3.62a) or (3.62b), can be expected. Otherwise, the polydisper-sity factor p will play a significant role. 

In creep measurements of polyacrylonitrile gels [78,82], the shear creep compliance J(t) behawd as JAt/tof/U + (t/to)"l where is the steady state compliance, the time constant to could be of the order of a minute, and n 0.75. This implies G co) (ko)" for coto 1 and S(q, t) for t to. We thus expect emergence of the power law (6.39) or more complicated transient decays in many cases. 

When reptation is used to develop a description of the linear viscoelasticity of polymer melts [5, 6], the same underlying hypothesis ismade, and the same phenomenological parameter Ng appears. Basically, to describe the relaxation after a step strain, for example, each chain is assumed to first reorganise inside its deformed tube, with a Rouse-like dynamics, and then to slowly return to isotropy, relaxing the deformed tube by reptation (see the paper by Montfort et al in this book). Along these lines, the plateau relaxation modulus, the steady state compliance and the zero shear viscosity should be respectively ... 

Another important point is that, when approaching Me, the tube consistency becomes weaker or in other words, the constraint release scaling law is modified and the rubbery plateau disappears whereas the steady-state compliance J decreEises. A self-consistent approach should predict that aroimd Me, the reptation modes would be gradually replaced by Rouse modes in order to describe the non entangled - entangled transition. 

The steady-state compliance follows the Rouse expression tmtil =5Mg. 

According to Doi-Edwards theory, the reptation of the long chains will occur in a tube whose diameter a veuies as Thus the number of monomers between entanglements will scale as < ) . Accordingly, the reptation time x (relation 3-14) should be proportional to (]) as a first approximation, the zero-shear viscosity tio and the steady-state compliance J should respectively scale as [Pg.133]

Therefore, for Gaussian molecules, the above parameters are functions of moments of the molecular weight distribution tiq a M,, and Jg a Mg.Mj+i/M. Otherwise, the mass dependence should be slightly different for qg and a large deviation from a combination of various average molecular weights is expected for the steady-state compliance. 

Therefore, whenever the temperature dependence of the steady-state compliance and the zero shear rate are known, the shift factors for viscoelastic liquids can be obtained directly from Eqs. (8.25) and (8.26). 

The double logarithmic plot of against molecular weight, shown in Figure 8.22 (34), indicates that the steady-state compliance is a linear function of M until a critical molecular weight M is reached, above which is nearly independent of molecular weight. Accordingly (32),... 

The steady-state compliance shows a strong dependence on the molecular heterodispersity. Thus the value of for a mixture of two fractions of the same polymer, one of low and the other of high molecular weight, may be up to 10 times as high as that of each component. This behavior can be explained by taking into account that 4 is the total recoverable deformation per unit of shear stress. The chains of high molecular weight have a very... 

Figure 8.22 Logarithmic plot of the steady state compliance, at — 30°C, versus weight-average molecular weight for poly(d5-isoprene). (From Ref. 34.)...

The strong effect of molecular chains on the viscoelastic behavior of polymeric solutions, even in the most dilute ones, is shown in Figure 8.24 (37). Here the recoverable compliance of a very dilute solution of polystyrene of weight-average molecular weight 860,000 in tri-m-tolyl phosphate is compared with that of the solvent. It is noteworthy that the value of the steady-state compliance for the solvent is 10 cm /dyn while that of the very dilute solution (Wpoi = 0.001) is nearly 10 cm /dyn. In other words, a very small fraction of the molecular chains are responsible for the fact that the steady-state compliance of the solution is more than 10 times that of the solvent. 

Both the steady-state compliance function, and the equilibrium compliance, Jg, can readily be obtained from the retardation spectrum. Actually, by taking the limit of Eq. (9.20) in the limit oo 0, the following relationship for Jg and Jg is obtained ... 

This equation indicates that the mean relaxation time is the product of two terminal viscoelastic functions, the zero shear rate viscosity and the steady-state compliance. The mean relaxation time can also be expressed in terms of the relaxation modulus by means of the expression... 

Though theory predicts the molecular weight independence of Jg for M3 > Mg( 6Mg), the theoretical values of are somewhat lower than the experimental ones. It should be pointed out that a certain degree of poly-dispersity may enhance the experimental values of the steady-state compliance of even so-called monodisperse systems. Finally, the theoretical... 

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