Mooney-Rivlin networks

The large deformability as shown in Figure 21.2, one of the main features of rubber, can be discussed in the category of continuum mechanics, which itself is complete theoretical framework. However, in the textbooks on rubber, we have to explain this feature with molecular theory. This would be the statistical mechanics of network structure where we encounter another serious pitfall and this is what we are concerned with in this chapter the assumption of affine deformation. The assumption is the core idea that appeared both in Gaussian network that treats infinitesimal deformation and in Mooney-Rivlin equation that treats large deformation. The microscopic deformation of a single polymer chain must be proportional to the macroscopic rubber deformation. However, the assumption is merely hypothesis and there is no experimental support. In summary, the theory of rubbery materials is built like a two-storied house of cards, without any experimental evidence on a single polymer chain entropic elasticity and affine deformation. 

The two network precursors and solvent (if present) were combined with 20 ppm catalyst and reacted under argon at 75°C to produce the desired networks. The sol fractions, ws, and equilibrium swelling ratio In benzene, V2m, of these networks were determined according to established procedures ( 1, 4. Equilibrium tensile stress-strain Isotherms were obtained at 25 C on dumbbell shaped specimens according to procedures described elsewhere (1, 4). The data were well correlated by linear regression to the empirical Mooney-Rivlin (6 ) relationship. The tensile behavior of the networks formed In solution was measured both on networks with the solvent present and on networks from which the oligomeric PEMS had been extracted. 

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. 

Networks were prepared in all cases using the amount of endlinking agent necessary to give a minimum Mc. Values of Mc were calculated from the Mooney-Rivlin elasticity coefficient Cj, determined from tensile stress-strain measurements (10),... 

Figure 11 shows plots according to equation(lO) of stress-strain data for triol-based polyester networks formed from the same reactants at three initial dilutions (0% solvent(bulk), 30% solvent and 65% solvent). Only the network from the most dilute reactions system has a strictly Gaussian stress-strain plot (C2 = 0), and the deviations from Gaussian behaviour shown by the other networks are not of the Mooney-Rivlin type. As indicated previously, they are more sensibly interpreted in terms of departures of the distribution of end-to-end vectors from Gaussian form. 

Figure 11. Mooney-Rivlin plot of stress-strain data (32) for three triol-based polyester networks prepared from sebacoyl chloride and LHT240 at various initial dilutions in diglyme as solvent. Conditions P100 is 0% solvent P130 is 30% solvent PI 65 is 65% solvent.

In this contribution, we report equilibrium modulus and sol fraction measurements on diepoxidet-monoepoxide-diamine networks and polyoxypropylene triol-diisocyanate networks and a comparison with calculated values. A practically zero (epoxides) or low (polyurethanes) Mooney-Rivlin constant C and a low and accounted for wastage of bonds in elastically inactive cycles are the advantages of the systems. Plots of reduced modulus against the gel fraction have been used, because they have been found to minimize the effect of EIC, incompleteness of the reaction, or possible errors in analytical characteristics (16-20). A full account of the work on epoxy and polyurethane networks including the statistical derivation of various structural parameters will be published separately elsewhere. 

The phantom network behaviour corresponding to volumeless chains which can freely interpenetrate one through the other and thus to unrestricted fluctuations of crosslinks should be approached in swollen systems or at high strains (proportionality to the Mooney-Rivlin constant C-j). For suppressed fluctuations of crosslinks, which then are displaced affinely with the strain, A for the small-strain modulus (equal to C1+C2) approaches unity. This situation should be characteristic of bulk systems. The constraints arising from interchain interactions important at low strains should be reflected in an increase of Aabove the phantom value and no extra Gee contribution to the modulus is necessary. The upper limit of the predicted equilibrium modulus corresponds therefore, A = 1. 

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. 

If material is neo-Hookean, its Mooney-Rivlin plot ought to give a horizontal line and hence yield C2 = 0. Thus one is tempted to consider that nonzero C2 must be associated in one way or another with the deviation of a given material from the idealized network model, and it is understandable why so many rubber scientists have concerned themselves with evaluating the C2 term from the Mooney-Rivlin plot of uniaxial extension data. However, the point is that a linear Mooney-Rivlin plot, if found experimentally, does not always warrant that its intercept and slope may be equated to 2(9879/,) and 2(91V/9/2), respectively. This fact is illustrated below with actual data on natural rubber (NR) and styrene-butadiene copolymer rubber (SBR). 

Even when the above complications are negligible or properly accounted for and when strain-induced crystallization is absent, the stress-strain curves for networks seldom conform to Eq. (7.3). The ratio //(a — 1/a2) generally decreases with elongation. An empirical extension of Eq. (7. IX the Mooney-Rivlin equation, has been used extensively to correlate experimental results ... 

It is well known that the equation of state of Eq. (28) based on the Gaussian statistics is only partially successful in representing experimental relationships tension-extension and fails to fit the experiments over a wide range of strain modes 29-33 34). The deviations from the Gaussian network behaviour may have various sources discussed by Dusek and Prins34). Therefore, phenomenological equations of state are often used. The most often used phenomenological equation of state for rubber elasticity is the Mooney-Rivlin equation 29 ,3-34>... 

In unfilled rubbers, which are not capable of strain-induced crystallization, the upturns on Mooney-Rivlin curves have shown to be absent 92 95). They disappear also in crystallizable rubbers at elevated temperatures and in the presence of solvents. On the other hand, the upturns do not appear for butadiene, nitrile and polyurethane rubbers if the limited chain extensibility function is introduced in the Mooney-Rivlin expression 97). Mark 92) has concluded that in the absence of selfreinforcement due to strain-induced crystallization or domains the rupture of the networks occurs long before the limited chain extensibility can be reached. 

Theories based on these concepts all have to take into account the phenomenology of the stress-strain behaviour of networks. In unilateral extension as well as compression one observes, even at moderate extension (1.1 deviations from the Gaussian behaviour, which can be empirically described by the so-called Mooney-Rivlin equation ... 

Some further remarks concerning the Mooney-Rivlin equation are in place (14, 112). In dry rubber networks Ca in extension is often of the same order of magnitude as Cx, so that we are by no means confronted with a minor correction. In unilateral compression C2 is almost zero, and perhaps slightly negative. The constant Cx increases with the crosslinking density and with the temperature the ratio C2/C( in extension seems... 

Figure 15.13 Slopes P=A/( k2- k i) versus the elastic modulus 2(Cj+C2) (Cj and C2 are the Mooney-Rivlin coefficients), in bimodal PDMS networks composed of a mixture of long (Mn = 25000 g.mol"1) and short chains (Mn = 3000 g.mol"1). Either long (capital letters) or short (small letters) chains are deuterated. The long chain volume fractions are 15% (d, D), 50% (m, M) and 75% (s, S)...

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 ... 

Figyre 3.16 Mooney-Rivlin plots [Eq. (3.38)] showing the effect of the temperature on stress-strain isotherms for model PDMS networks (15,16). The filled circles represent the reversibility of the elastic measurements, and the vertical lines locate the fracture points. (From Ref. 15.)... 

For classical models, the Mooney-Rivlin coefficients are 2Ci=G and 2 = 0. However, experimental data plotted in Fig. 7.13, in the form suggested by Eq. (7.59), show that C2>0. In this Mooney-Rivlin plot, the stress divided by the prediction of the classical models is plotted as a function of the reciprocal deformation 1/A. The predictions of the affine, phantom, and Edwards tube network models correspond to horizontal lines on the Mooney-Rivlin plot (C2 = 0). Experimental data on uniaxial... 

Mooney-Rivlin plots for uniaxial tension data on three networks prepared from radiation-crosslinking a linear polybutadiene melt with------------------... 

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