Network reduced stress

The storage modulus (G ) was recorded at a frequency of IHz under 0.015 strain amplitude until stabilization of the protein network. In order to reduce stress in the sample, G recording started just before the gelation time which corresponds to the time at which G deviated from the baseline. Data were collected and rheological parameters were calculated using Carri-Med 50 software. For each system, the experiments were performed in triplicate. 

Comparison with Statistical Theory at Moderate Strains. So far we have shown, that a transition between the two limiting classical theories, i.e. affine theory and phantom theory, is possible by a suitable choice of the network microstructure. This argument goes beyond the revised theory by Ronca and Allegra and by Flory, which predicts such a transition as a result of increasing strain, thus explaining the experimentally observed strain dependence of the reduced stress. 

Classical molecular theories of rubber elasticity (7, 8) lead to an elastic equation of state which predicts the reduced stress to be constant over the entire range of uniaxial deformation. To explain this deviation between the classical theories and reality. Flory (9) and Ronca and Allegra (10) have separately proposed a new model based on the hypothesis that in a real network, the fluctuations of a junction about its mean position may may be significantly impeded by interactions with chains emanating from spatially, but not topologically, neighboring junctions. Thus, the junctions in a real network are more constrained than those in a phantom network. The elastic force is taken to be the sum of two contributions (9) ... 

Number-average molecular weights are Mn = 660 and 18,500 g/ mol, respectively (15,). Measurements were carried out on the unswollen networks, in elongation at 25°C. Data plotted as suggested by Mooney-Rivlin representation of reduced stress or modulus (Eq. 2). Short extensions of the linear portions of the isotherms locate the values of a at which upturn in [/ ] first becomes discernible. Linear portions of the isotherms were located by least-squares analysis. Each curve is labelled with mol percent of short chains in network structure. Vertical dotted lines indicate rupture points. Key O, results obtained using a series of increasing values of elongation 0, results obtained out of sequence to test for reversibility. 

The results of stress-strain measurements can be summarized as follows (1) the reduced stress S (A- A ) (Ais the extension ratio) is practically independent of strain so that the Mooney-Rivlin constant C2 is practically zero for dry as well as swollen samples (C2/C1=0 0.05) (2) the values of G are practically the same whether obtained on dry or swollen samples (3) assuming that Gee=0, the data are compatible with the chemical contribution and A 1 (4) the difference between the phantom network dependence with the value of A given by Eq.(4) and the experimental moduli fits well the theoretical dependence of G e in Eq.(2) or (3). The proportionality constant in G for series of networks with s equal to 0, 0.2, 0.33, and 0. Ewas practically the same -(8.2, 6.3, 8.8, and 8.5)x10-4 mol/cm with the average value 7.95x10 mol/cm. Results (1) and (2) suggest that phantom network behavior has been reached, but the result(3) is contrary to that. Either the constraints do survive also in the swollen and stressed states, or we have to consider an extra contribution due to the incrossability of "phantom" chains. The latter explanation is somewhat supported by the constancy of in Eq.(2) for a series of samples of different composition. 

Other interesting features of elastomeric networks can be revealed using the plots of the reduced stress, crred = /( — -2) against inverse extension ratio 1. This can be analyzed from the stress-strain behavior described by a phenomenological expression suggested by Mooney [78] and Rivlin and Saunders [79] ... 

The stress-strain curve for unfilled NR exhibits a large increase in stress at higher deformations. NR displays, due to its uniform microstructure, a very unique important characteristic, that is, the ability to crystallise under strain, a phenomenon known as strain-induced crystallization. This phenomenon is responsible for the large and abrupt increase in the reduced stress observed at higher deformation corresponding, in fact, to a self-toughening of the elastomer because the crystallites act as additional cross-links in the network. This process can be better visualized by using a Mooney-Rivlin representation, based on the so-called Mooney-Rivlin equation ... 

Mackwell et al. (1985) found that when specimens that had been deformed under anhydrous conditions were subsequently further deformed under wet conditions, there was a significant change in microstructure. TEM observations revealed enhanced formation of dislocation walls, despite the reduced stress levels. This observation was interpreted as due to enhanced dislocation climb under wet conditions. However, the two walls illustrated by Mackwell et al. (1985) could be interpreted as healed or partly healed fractures. One wall consists of a very irregular network of dislocations with many bubbles, particularly at dislocation intersections. 

Both the affine and the phantom network models predict that the reduced stress, [/ ], measured in uniaxial deformation is independent of the deformation ratio. However, it... 

FIGURE 29.5. Mooney-Rivlin reduced stress plot showing comparison of experimental data with modified constrained chain model (MCC) predictions for dry (o) and swollen ( ) natural rubber networks [112,117]. Swelling agent n-Decane. continuous lines are theoretical curves calculated with paremeters /cT/l/o = 0.17MPa and kq =2.0. 

Figure 7.1 schematically shows the preparation of networks by cross linking in solution followed by removal of the solvent. Success in obtaining elastomers with fewer entanglements is supported by the observation that such networks come to elastic equilibrium much more rapidly than elastomers cross linked in the dry state. Table 7.1 shows results on PDMS networks cross linked in solution by means of y radiation. - Note the continual decrease in the time required to reach elastic equilibrium, and in the extent of stress relaxation as measured by the ratio of equilibrium to initial values of the reduced stress, [f ], upon decrease in the volume... 

Schematic stress-strain isotherms in elongation for a unimodal elastomer in the Mooney-Rivlin representation of modulus against reciprocal elongation. The isotherms are represented as the dependence of the reduced stress ([f ] = f /(a - on reciprocal elongation. (f = f/A, f = elastic force, A = undeformed area, a = elongation). The top three are for a crystallizable network curve A for a relatively low temperature, B for an increased temperature, and C for the introduction of a swelling diluent. Isotherm D is for an unswollen unimodal network that is inherently noncrystallizable.

Schematic recovery curves for constant strain and stress-free conditions for thermoplastics and polymeric networks are shown in Fig. 9. Recovery of a thermoplastic SMP under constant strain initially leads to an increase of cr until is reached. At higher temperatures a drop in stress can be observed, where the softening of the polymer dominates, which is caused by an increase in mobility of the chain segments. In SMP networks the stress remains constant above and is only reduced after cooling the sample again.

Rg. 4.15 Reduced stresses in the swollen (upper line) and unswollen (lower line) states. Experimental data lie in-between the limits predicted by the affine and phantom network theories. The tension reduces on swelling. (Reprinted with permission from Ref. [5], Chap. 8.)... 

Figure 10.7 (8) illustrates the stress relaxation of a poly(dimethyl siloxane) network, silicone rubber, in the presence of dry nitrogen. The reduced stress, o(t)/(T(0),is plotted,so that under the initial conditions its value is always unity. Since the theory of rubber elasticity holds (Chapter 9), what is really measured is the fractional decrease in effective network chain segments. The bond interchange reaction of equation (10.2) provides the chemical basis of the process. While the rate of the relaxation increases with temperature, the lines remain straight, suggesting that equation (10.2) can be treated as the sole reaction of importance. 

For both the phantom and the affine networks, the reduced stress is calculated to be independent of deformation. However, stress-strain measurements carried out in uniaxial extension of dry and swollen networks have revealed departures from these predictions of simple models 5. These observations then gave rise to phenomenological equations like the Mooney-Rivlin expression, i.e. 

Experimental results indicate that the response to deformation of a network generally falls between the affine and phantom limits [31-34]. At low deformations, chain-junction entangling suppresses the fluctuations of the junctions and the deformation is relatively close to the affine limit. This is illustrated in Fig. 1.8, which shows schematically some of the results of the constrained-junction theory based on this qualitative idea [32-34]. In the case of the two limits, the affine deformation and the non-affine deformation in the phantom-network limit, the reduced stress should be independent of a. Because of junction fluctuations, the value for the... 

It is interesting to compare eq. (3.54) with the expressions obtained from the statistical theories (Fig. 3.20). According to both the affine network model and the phantom network model of James and Guth, the reduced stress remains constant and independent of strain, which is not the case for the Mooney-Rivlin equation. 

Figure 12.2. Variation of the reduced stress versus the reverse of elongation for various models of eiastomeric networks.

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