Plateau compliance

The plateau compliance is characteristic of rubbery behavior where chain entanglements play the role of effective crosslinks. 

The early attempts to interpret the dynamic mechanical behaviour in structural terms include that of Smith et al. where the plateau modulus was correlated with the fraction of non-crystalline material f, determined by NMR. Plots of the plateau compliances at —60 °C and —160 °C as a function of f suggested a modified Takay-anagi series model, with a constant amount of non-crystalline material in parallel with the simple series model. The modd showed good internal consistency, with values for the compliances of the non-ciystalline regions which were acceptable in physical terms. 

Figure 3 Variation of the plateau compliance JN° of gluten (Olympic x Gabo cross line -// 7 + 18/-) with temperature in water filled symbols) and in deuterium dioxide (empty symbols). J N was obtained from the fit of the Cole-Cole functions to the dynamic measurements data...

Applying the kinetic theory of rubber-like elasticity (62,109) to the entanglement network one can determine the molecular weight between entanglements Me from the plateau compliance Jn or plateau modulus Gn (62) ... 

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 ... 

Steady-State Compliances and Ratios to Plateau Compliances, Narrow Distribution Polymers, M M c ... 

In branched polymers, the plateau compliance is often not well defined even for narrow molecular weight distribution, there is no obvious plateau. - This is probably related to the suppression of reptation, which forces each molecule to relax by different mechanisms with a broad spectrum of relaxation times. 

The plateau compliance J% or its reciprocal, the modulus G%, can be obtained from stress relaxation, creep, or dynamic data by integrations such as equations 3 to 5 of Chapter 13 in order to determine its concentration dependence without complication of time scale shifts. Examples are shown in Fig. 17-14 for concentrated solutions of poly(methyl methacrylate) and m-polyisoprene. The... 

This type of plot magnifles the pre-steady-state behavior and new features become visible, in particular, the plateau compliance /, which is the reciprocal of the plateau modulus, i.e.. 

The upswing in compliance from the rubbery plateau marks the onset of viscous flow. In this final stage the slope of the lines (the broken lines in Fig. 3.12) is unity, which means that the compliance increases linearly with time. 

Lowering of the rubbery plateau modulus increases the compliance of the polymer making faster wet-out of a substrate possible. As a result, the PSAs show more aggressive tack properties. Provided the surface energy of the substrate allows for complete polymer wetting, a PSA with improved quick-stick and faster adhesion build will be obtained. 

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 25 PHR-DDS N-5208 epoxy sample covers a broader range of mole-moisture per mole-nitrogen values In Figure 4. However, at an abscissa value of 0.8, the differential loss compliance area does not appear saturated. The higher functionality of the N-5208 components result In a cured network with fewer tertiary amine links (6). This, In conjunction with the presence of the DDS sulfone groups, can be used to rationalize the lack of a plateau In the differential area curves of the 25 PHR-DDS epoxy data. 

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 ... 

For a given polymer, the viscoelastic curves (either moduli or compliances) obtained at different temperatures in the plateau and terminal regions are simply affine in the frequency (or time) scale, in a double logarithmic plot. 

The viscosity, plateau modulus, limiting compliance and maximum (terminal) relaxation time derived firom the basic D-E model are power laws of the molecular weight M ... 

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 small changes taking place in the compliance function with time at the plateau also obey Andrade s equation for creep, that is (26-28),... 

In the glass-like zone, the values of the creep compliance function seem to be independent of the concentration however, the changes that take place in the values of J t) in the transition zone are larger the lower the concentration. The length of the plateau increases with the concentration, and the plateau and terminal zones merge into a single region at low concentrations. The location of the isotherms on the time scale is shifted to shorter chains as the concentration decreases. 

Figure 8.26 Double logarithmic plots showing the concentration dependence of both the creep compliance and the relaxation modulus at the plateau. (From Ref. 8.)...

---