Modulus plateau

Owing to the fact that the tube diameter is always larger than the correlation length (fl 0, the entanglement strand is a random walk of correlation volumes in any solvent  

Dilution effect on the plateau modulus of linear polymers, Filled diamonds are polystyrene in cyclohexane at 34.5 °C (0-solvent), open squares are polystyrene in benzene at 25 °C (good solvent), filled circles are polybutadiene in dioctylphthalate at 25°C (near  

0-solvent) and open triangles are polybutadiene in phenyloctane (good solvent), PS data from M. Adam and M. Delsanti, J. Phys. France 44, 1185 (1983) 45, 1513 (1984). PB data from R. H. Colby et al, Macromolecules 24, 3873 (1991). 

Topological constraints do not influence polymer motion on length scales smaller than the size of an entanglement strand. In entangled polymer solutions, chain sections with end-to-end distance shorter than the tube 

On length scales larger than the correlation length but smaller than the tube diameter a, hydrodynamic interactions are screened, and topological interactions are unimportant. Polymer motion on these length scales is described by the Rouse model. The relaxation time Tg of an entanglement strand of monomers is that of a Rouse chain of N jg correlation volumes [Eg. (8.76)]  

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At still longer times a more or less pronounced plateau is encountered. The value of the plateau modulus is on the order of 10 N m", comparable to the effect predicted for cross-linked elastomers in Sec. 3.4. This region is called the rubbery plateau and the sample appears elastic when observed in this time frame. 

Crosslinking the PSA will increase the solvent resistance of the material and it will also have a significant effect on the rubbery plateau modulus of the polymer. Fig. 8 shows the effect of increasing amounts of a multifunctional az.iridine crosslinker, such as CX-100 (available from Avecia, Blackley, Manchester, UK) on the rheology of an acrylic polymer containing 10% acrylic acid. The amounts of crosslinker are based by weight on the dry weight of the PSA polymer. 

Increasing the amount of crosslinker extends the plateau modulus to higher and higher temperatures, eventually eliminating the flow of the polymer. The effect on the glass transition is minimal. 

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. 

Similar to the tackifiers discussed earlier, plasticizers have a very dramatic softening effect on the rubbery plateau modulus of the PSA. For this reason, high levels of plasticizers have to be avoided to maintain good cohesive strength in the adhesive, especially at elevated temperatures. Indeed, if high cohesive strength is desired, the amount of plasticizer used in a PSA is typically kept to a minimum, if used at all. 

Investigation of the linear viscoelastic properties of SDIBS with branch MWs exceeding the critical entanglement MW of PIB (about -7000 g/mol ) revealed that both the viscosity and the length of the entanglement plateau scaled with B rather than with the length of the branches, a distinctively different behavior than that of star-branched PIBs. However, the magnitude of the plateau modulus and the temperature dependence of the terminal zone shift factors were found to... 

For highly concentrated solutions, Eq. (20) can be simplified, due to the fact that in such a case the plateau modulus G p is Mw independent. Under these conditions it is clear that E is solely a function of... 

So far, the existence of a well-defined entanglement length in dense polymer systems has been inferred indirectly from macroscopic experiments like measurements of the plateau modulus. However, its direct microscopic observation remained impossible. The difficulty in directly evaluating the entanglement... 

In the Doi-Edwards theory the plateau modulus and the tube diameter are related according to Eq. (40). Inserting Eq. (40) into (52) we finally obtain... 

The Gge contribution of the form given Eq.(2) represents the simplest form of permanent interchain interactions. The value of Gee at Teg=1 and w =1, i.e. the Gge contribution of a perfect network, has been assumed equal to the plateau modulus of the corresponding linear polymer (10,15,23). This assumption has not always been confirmed and, therefore, for the purpose of this work we prefer to consider g of g" as proportionality constants. 

Comments on Calculated Data. In several studies (13,18,19), G ax has been found to equal, or possibly be somewhat less than, the plateau modulus, G j, of a high molecular weight polymer whose chemical composition is the same as that of the network chains. Although G j for amorphous PPO has not been reported, it can be estimated from Zc, the number of chain atoms per molecule above which the viscosity increases approximately with the 3.4 power of Z. This quantity has been reported (25,26) to be about 400. As the chain atoms between entanglements is commonly about Zc/2, it follows that the molecular weight between entanglement loci is about 3900, and thus G j [ = (p/Me)RT] is about 0.65 MPa at 30°C. 

Since the excellent work of Moore and Watson (6, who cross-linked natural rubber with t-butylperoxide, most workers have assumed that physical cross-links contribute to the equilibrium elastic properties of cross-linked elastomers. This idea seems to be fully confirmed in work by Graessley and co-workers who used the Langley method on radiation cross-linked polybutadiene (.7) and ethylene-propylene copolymer (8) to study trapped entanglements. Two-network results on 1,2-polybutadiene (9.10) also indicate that the equilibrium elastic contribution from chain entangling at high degrees of cross-linking is quantitatively equal to the pseudoequilibrium rubber plateau modulus (1 1.) of the uncross-linked polymer. 

Unfortunately, the method is based on a fairly large nunber of assumptions. If we want to relate GN to the pseudo-equilibrium rubber plateau modulus, G , and to the effect of chain entangling in ordinary networks produced by cross-linking in the unstrained state, the following assumptions are required ... 

Figure 3. Modulus contributions from chemical cross-links (Cx, filled triangles) and from chain entangling (Gx, unfilled symbols) plotted against the extension ratio during cross-linking, A0, for 1,2-polybutadiene. Key O, GN, equibiaxial extension , G.v, pure shear A, Gx, simple extension Gx°, pseudo-equilibrium rubber plateau modulus for a polybutadiene with a similar microstructure. See Ref. 10.

The two-network method has been carefully examined. All the previous two-network results were obtained in simple extension for which the Gaussian composite network theory was found to be inadequate. Results obtained on composite networks of 1,2-polybutadiene for three different types of strain, namely equibiaxial extension, pure shear, and simple extension, are discussed in the present paper. The Gaussian composite network elastic free energy relation is found to be adequate in equibiaxial extension and possibly pure shear. Extrapolation to zero strain gives the same result for all three types of strain The contribution from chain entangling at elastic equilibrium is found to be approximately equal to the pseudo-equilibrium rubber plateau modulus and about three times larger than the contribution from chemical cross-links. 

The adjustable parameters, G (o), t0, m, AG, a and p were determined by a least-squares analysis. In physical terms G (o) Is the equilibrium modulus, AG represents the Increment to the entanglement plateau modulus, t0 Is a characteristic time, related, for the last three equations, to the frequency In the point of Inflection, and the exponents m, a and p relate to the slope at the characteristic frequency. 

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