Steady-state compliance theory

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. 

The polydispersity factor p is evaluated with the aid of any of the well-known viscosity-molecular weight relationships. From eq. (3.60) a proportionality of the intrinsic viscosity to the half power of the molecular weight is expected, as this theory holds for 0-solvents. However, based on the conclusion of Section 3.5, viz. that the reduced steady-state compliance of a monodisperse polymer is insensitive to the excluded... 

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]

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... 

Fig. 10.11 Comparison of the steady-state compliance data, x pRT, of nearly monodisperse polystyrene samples ( and from Ref. 18 A from Ref. 33) and those calculated from Eq. (9.25) (solid line 1), from the Doi-Edwards theory (the dashed line), from the Rouse theory (the dotted line), and calculated numerically from substituting Eq. (9.19) into Eq. (4.63) with K jK = 1 (line 2), and K jK = 3.3 (line 3).

We can conclude that the ERT has accurately explained the molecular-weight dependence of the zero-shear viscosity and the steady-state compliance and their respective transition points Me and M. This success is indeed the logical consequence of the success of the theory in analyzing the G t) curves of the studied samples, a vitally important aspect of which is the molecular-weight independence of the frictional factor K. Prom the analysis of the G t) curves, it is revealed that entanglements exist between Mg and Me- This point will be further confirmed by the... 

Fig. 13. Steady state compliance oflinear and branched polyisoprenes in tetradecane (0.33 g/ml) at 25°. Symbols as in Fig. 8, Ref. Eteshed lines are values calculated from Rouseflam Theory...

For the restricted case of low molecular weights and no coupling entanglements, the viscoelastic properties of star-branched undiluted polymers can be described by a special case of the Zimm-Kilb theory o in which there is no hydrodynamic interaction. Calculations were made by Ham i by use of a method which is somewhat different from that of Rouse but yields the same results for unbranched molecules. Stars with arms of unequal length were included. For such a branched molecule, the terminal relaxation time ti, the viscosity r/o, and the steady-state compliance are always smaller than for an unbranched molecule of the same molecular weight the more branches and the more nearly equal their lengths, the... 

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 ... 

In the course of tensile creep, the form of the time dependence of strain (as expressed by the stretch ratio X, for example) depends on the magnitude of tensile stress at high stresses." " Recovery is considerably more rapid than would be predicted from the Boltzmann superposition principle, as illustrated in Fig. 13-23 for polyisobutylene of high molecular weight. " The course of recovery is predicted successfully by the theory of Bernstein, Kearsley, and Zapas. 2 - 22 -pije stress-dependent recoverable steady-state compliance D = which is equal to Z) at low stresses, decreases with increasing Ot- This effect, moderate when the tensile strain e is defined as X — 1, is more pronounced when it is replaced by the Hencky strain, defined as In X. The stress dependence of steady-state compliance in shear will be discussed in Chapter 17. The reader is referred to the review by Petrie" for more details. 

The steady-state compliance, as expressed by the theories of Graessley or Doi and Edwards for undiluted polymers (equations 52 and 56 of Chapter 10), is proportional to Me/p and independent of M when M MS- For concentrated solutions, p must be replaced by c = pv2, and, if Me — M v as discussed in Section B1 above, the concentration dependence of 7° under conditions of high entanglement becomes... 

In this section, we present the molecular theory for the linear dynamic viscoelasticity of miscible polymer blends by Han and Kim (1989a, 1989b), which is based on the concept of the tube model presented in Chapter 4. Specifically, the reptation of two primitive chains of dissimilar chemical structures under an external potential will be considered, and the expressions for the linear viscoelastic properties of miscible polymer blends will be presented. We will first present the expressions for zero-shear viscosity ob. dynamic storage and loss moduli G co) and G " co), and steady-state compliance J° for binary miscible blends of monodisperse, entangled flexible homopolymers and then consider the effect of polydispersity. There are a few other molecular theories reported... 

According to the theory of linear elastico-viscous behaviour (47) the steady-state shear viscosity t] and the steady-state shear compliance Je depend in the following way on the shear relaxation modulus G (t), where t is here the time of the relaxation experiment ... 

Unfortunately, Fixman has not yet given a value for the reduced steady-state shear compliance. However, from a comparison of eqs. (3.60a), (3.64) and (3.66) the impression is obtained that the theory of Ptitsyn and Eizner overestimates the influence of the excluded volume on 0 and JeR. As will be shown in the experimental section of this chapter, this impression is supported by flow birefringence measurements on solutions in 0- solvents and in good solvents. 

Fig. 23. Normalized reciprocal steady-state recoverable compliance Je,max/Je for three polymers, poly(dimethyl siloxane), PIB, and PS, versus the reduced temperature T/Tg Tg is the glass temperature and the normalized compliance

In addition to timescale shifts with temperature, the magnitude of the compliance or modulus can change. The kinetic theory of rubber-like elasticity suggests that the entropically based contribution of the modulus to the viscoelastic response should increase in direct proportion to the absolute temperature. Correspondingly, the reciprocal of the steady-state recoverable compliance should be directly proportional to the absolute temperature. This is true at temperatures that are greater than 2Tg, but, between 1.2Tg and 2Tg, the steady-state recoverable compliance Js is essentially independent of temperature. At still lower temperatures a strong decrease of Js, is seen [51]. 

Masao Doi and Sam F. Edwards (1986) developed a theory on the basis of de Genne s reptation concept relating the mechanical properties of the concentrated polymer liquids and molar mass. They assumed that reptation was also the predominant mechanism for motion of entangled polymer chains in the absence of a permanent network. Using rubber elasticity theory, Doi and Edwards calculated the stress carried by individual chains in an ensemble of monodisperse entangled linear polymer chains after the application of a step strain. The subsequent relaxation of stress was then calculated under the assumption that reptation was the only mechanism for stress release. This led to an equation for the shear relaxation modulus, G t), in the terminal region. From G(t), the following expressions for the plateau modulus, the zero-shear-rate viscosity and the steady-state recoverable compliance are obtained ... 

The Curtiss-Bird theory has been extended to include polydispersity effects extensive data comparisons for the steady-state shear compliance and various nonlinear rheological properties further support the inclusion of the link-tension coefficient in the theory. 

---