Stress-relaxation two-network

A new stress-relaxation two-network method is used for a more direct measurement of the equilibrium elastic contribution of chain entangling in highly cross-linked 1,2-polybutadiene. The new method shows clearly, without the need of any theory, that the equilibrium contribution is equal to the non-equilibrium stress-relaxation modulus of the uncross-linked polymer immediately prior to cross-linking. The new method also directly confirms six of the eight assumptions required for the original two-network method. 

A8. The Helmholtz elastic free energy relation of the composite network contains a separate term for each of the two networks as in eq. 5. However, the precise mathematical form of the strain dependence is not critical at small deformations. Although all the assumptions seem to be reasonably fulfilled, a simpler method, which would require fewer assumptions, would obviously be desirable. A simpler method can be used if we just want to compare the equilibrium contribution from chain engangling in the cross-linked polymer to the stress-relaxation modulus of the uncross-linked polymer. The new method is described in Part 3. 

Because of the interaction of the two complicated and not well-understood fields, turbulent flow and non-Newtonian fluids, understanding of DR mechanism(s) is still quite limited. Cates and coworkers (for example, Refs. " ) and a number of other investigators have done theoretical studies of the dynamics of self-assemblies of worm-like micelles. Because these so-called living polymers are subject to reversible scission and recombination, their relaxation behavior differs from reptating polymer chains. An additional form of stress relaxation is provided by continuous breaking and repair of the micellar chains. Thus, stress relaxation in micellar networks occurs through a combination of reptation and breaking. For rapid scission kinetics, linear viscoelastic (Maxwell) behavior is predicted and is observed for some surfactant systems at low frequencies. In many cationic surfactant systems, however, the observed behavior in Cole-Cole plots does not fit the Maxwell model. 

The rheological behavior of these materials is still far from being fully understood but relationships between their rheology and the degree of exfoliation of the nanoparticles have been reported [73]. An increase in the steady shear flow viscosity with the clay content has been reported for most systems [62, 74], while in some cases, viscosity decreases with low clay loading [46, 75]. Another important characteristic of exfoliated nanocomposites is the loss of the complex viscosity Newtonian plateau in oscillatory shear flow [76-80]. Transient experiments have also been used to study the rheological response of polymer nanocomposites. The degree of exfoliation is associated with the amplitude of stress overshoots in start-up experiment [81]. Two main modes of relaxation have been observed in the stress relaxation (step shear) test, namely, a fast mode associated with the polymer matrix and a slow mode associated with the polymer-clay network [60]. The presence of a clay-polymer network has also been evidenced by Cole-Cole plots [82]. 

The sorption of hydrocarbons proved to be a strongly exothermic process. This is not surprising because both the adsorption (pentane-polystyrene dispersion interactions) and the relaxation of strong inner stresses of the network result in heat generation. The temperature increase of the packing affects the sorption process in two different ways. First, it results in a considerable acceleration of the sorption process due to the facilitated diffusion of the sorbate molecules into the bead interior (the effective diffusion coefficient increases by 1 order of magnitude with the temperature... 

Results of dynamic mechanical measurements are shown in Fig. 42 none of the gels show a real equilibrium plateau for G, so a real permanent network is not obtained. Moreover, the gels show thixotropic (two phase) behaviour, which follows from the observations that the gels are liable to a continuous stress relaxation upon applying a constant deformation and that the linearity of the dynamic moduli already disappears at strains of less than 1% the non-linear response, however, is almost instantaneous and, by turning back into the linear region, the initial values are quickly regained. These phenomena are typical for two-phase systems and especially for thixotropic ones. 

In the experiments, two main types of time-dependent flows have been studied start-up flows and stress relaxation. In the start-up flow experiments, shear flows with constant shear rates and elongational flows with constant elongational rates are started in the system in equilibrium under no external force, and the time-dependent stress build-up in the system is measured. In the stress relaxation experiments, constant deformations are applied to or removed from the system, and the time-dependent relaxation of the stress is measured. In this section, we study these two types within the framework of transient network theory. 

Chapter 9 presents the transient network theory of associating polymer solutions, which is the other one of the two major theories treated in this book. It studies the dynamic and rheological flow properties of structured solutions from a molecular point of view. Thus, linear complex modulus, nonlinear stationary viscosity, start-up flows, and stress relaxation in reversible polymer networks are studied in detail. 

The relative magnitudes of C- and Cpj, are expressed by the ratio = Cp / C- + Cp ) which is aetermined experimentally from nonlinear stress relaxation experiments on the uncross-linked polymer. The contributions to the true stress a of the dual network at any stretch ratio X from the two individual networks are then... 

The mathematics of stress-relaxation measurements have been extensively developed and reviewed in the literature (e.g. Dunn and Scanlan, 1963). They will not therefore be elaborated here but it is to be noted that in a network in which chain ends and physical entanglements can be ignored a scission reaction in a chain between cross-links will reduce the number of stress-supporting chains by one whilst if there is cross-link scission the number of stress-supporting chains is reduced by two (Fig. 9.3). 

The extensive stress relaxation experiments of Tobolsky on uncross-linked polymers of high molecular weight similarly showed two stages of relaxation (like those in Examples III and IV of Fig. 2-2) with a period in the time scale where the stress relaxed very slowly, leading to the concept of an entangled network structure. 

If a mixture of two polymeric species which are almost identical chemically except that only one of them carries cross-linkable reactive groups is subjected to cross-linking, a network is formed through which are threaded unattached macromolecules. From viscoelastic measurements, the motions of these molecules through the network can be deduced, as illustrated in Fig. 14-11 for stress relaxation of ethylene-propylene terpolymer networks containing 25% of unattached linear copolymer with essentially the same chemical composition. A network containing no unattached species undergoes very little relaxation in the time scale covered. 

If this model is correct then we should have a maximum in relaxation times in the region between the two transitions. On the lower temperature side of the maximum, the relaxation times go up because as temperature is increased there is an opportunity for more and more long range uncoiling between entanglement network points. When the temperature is further increased the network entanglement points begin to "melt out" and the stress relaxation becomes mote rapid. 

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